PyMC Documentation Release 2.3.5 Christopher J. Fonnesbeck August 20, 2015 Contents 1 2 3 4 5 Introduction 1.1 Purpose . . . . . . . . . . . . 1.2 Features . . . . . . . . . . . . 1.3 What’s new in version 2 . . . 1.4 Usage . . . . . . . . . . . . . 1.5 History . . . . . . . . . . . . 1.6 Relationship to other packages 1.7 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 3 3 4 4 5 5 5 Installation 2.1 Dependencies . . . . . . . . . . 2.2 Installation using EasyInstall . . 2.3 Installing from pre-built binaries 2.4 Compiling the source code . . . 2.5 Installing from GitHub . . . . . 2.6 Running the test suite . . . . . 2.7 Bugs and feature requests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 7 8 8 8 9 9 9 Tutorial 3.1 An example statistical model . . . . . . 3.2 Two types of variables . . . . . . . . . 3.3 Parents and children . . . . . . . . . . 3.4 Variables’ values and log-probabilities . 3.5 Fitting the model with MCMC . . . . . 3.6 Fine-tuning the MCMC algorithm . . . 3.7 Beyond the basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 11 12 14 14 17 22 22 Building models 4.1 The Stochastic class . . . . . . 4.2 Data . . . . . . . . . . . . . . . 4.3 The Deterministic class . . . . 4.4 Containers . . . . . . . . . . . 4.5 The Potential class . . . . . . . 4.6 Graphing models . . . . . . . . 4.7 Class LazyFunction and caching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 26 26 28 29 31 31 Fitting Models 5.1 Creating models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . . i 5.2 5.3 5.4 5.5 5.6 5.7 5.8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 37 37 38 39 40 43 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 45 46 46 49 7 Model checking and diagnostics 7.1 Convergence Diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Autocorrelation Plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Goodness of Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 51 59 60 8 Extending PyMC 8.1 Nonstandard Stochastics . . . . . . . . . . . . . . . . . . . . . . . . 8.2 User-defined step methods . . . . . . . . . . . . . . . . . . . . . . . 8.3 New fitting algorithms . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 A second warning: Don’t update stochastic variables’ values in-place . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 65 65 69 70 Probability distributions 9.1 Discrete distributions . . . . . . . . . 9.2 Continuous distributions . . . . . . . 9.3 Multivariate discrete distributions . . 9.4 Multivariate continuous distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 71 71 71 71 6 9 The Model class . . . . . . . . . . . . . . . Maximum a posteriori estimates . . . . . . . Normal approximations . . . . . . . . . . . Markov chain Monte Carlo: the MCMC class The Sampler class . . . . . . . . . . . . . . Step methods . . . . . . . . . . . . . . . . . Gibbs step methods . . . . . . . . . . . . . . Saving and managing sampling results 6.1 Accessing Sampled Data . . . . . 6.2 Saving Data to Disk . . . . . . . 6.3 Reloading a Database . . . . . . 6.4 Writing a New Backend . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Conclusion 73 11 Acknowledgements 75 12 Appendix: Markov Chain Monte Carlo 12.1 Monte Carlo Methods in Bayesian Analysis . . . 12.2 Markov Chains . . . . . . . . . . . . . . . . . . 12.3 Why MCMC Works: Reversible Markov Chains 12.4 Gibbs Sampling . . . . . . . . . . . . . . . . . 12.5 The Metropolis-Hastings Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 80 81 82 82 13 List of References 85 14 Indices and tables 87 Bibliography 89 Python Module Index 91 ii PyMC Documentation, Release 2.3.5 Contents: Contents 1 PyMC Documentation, Release 2.3.5 2 Contents CHAPTER 1 Introduction Date 3 September 2013 Authors Chris Fonnesbeck, Anand Patil, David Huard, John Salvatier Contact [email protected] Web site http://github.com/pymc-devs/pymc Copyright This document has been placed in the public domain. License PyMC is released under the Academic Free License. Version 2.3 1.1 Purpose PyMC is a python module that implements Bayesian statistical models and fitting algorithms, including Markov chain Monte Carlo. Its flexibility and extensibility make it applicable to a large suite of problems. Along with core sampling functionality, PyMC includes methods for summarizing output, plotting, goodness-of-fit and convergence diagnostics. 1.2 Features PyMC provides functionalities to make Bayesian analysis as painless as possible. Here is a short list of some of its features: • Fits Bayesian statistical models with Markov chain Monte Carlo and other algorithms. • Includes a large suite of well-documented statistical distributions. • Uses NumPy for numerics wherever possible. • Includes a module for modeling Gaussian processes. • Sampling loops can be paused and tuned manually, or saved and restarted later. • Creates summaries including tables and plots. • Traces can be saved to the disk as plain text, Python pickles, SQLite or MySQL database, or hdf5 archives. • Several convergence diagnostics are available. • Extensible: easily incorporates custom step methods and unusual probability distributions. • MCMC loops can be embedded in larger programs, and results can be analyzed with the full power of Python. 3 PyMC Documentation, Release 2.3.5 1.3 What’s new in version 2 This second version of PyMC benefits from a major rewrite effort. Substantial improvements in code extensibility, user interface as well as in raw performance have been achieved. Most notably, the PyMC 2 series provides: • New flexible object model and syntax (not backward-compatible). • Reduced redundant computations: only relevant log-probability terms are computed, and these are cached. • Optimized probability distributions. • New adaptive blocked Metropolis step method. • New slice sampler method. • Much more! 1.4 Usage First, define your model in a file, say mymodel.py (with comments, of course!): # Import relevant modules import pymc import numpy as np # Some data n = 5*np.ones(4,dtype=int) x = np.array([-.86,-.3,-.05,.73]) # Priors on unknown parameters alpha = pymc.Normal('alpha',mu=0,tau=.01) beta = pymc.Normal('beta',mu=0,tau=.01) # Arbitrary deterministic function of parameters @pymc.deterministic def theta(a=alpha, b=beta): """theta = logit^{-1}(a+b)""" return pymc.invlogit(a+b*x) # Binomial likelihood for data d = pymc.Binomial('d', n=n, p=theta, value=np.array([0.,1.,3.,5.]),\ observed=True) Save this file, then from a python shell (or another file in the same directory), call: import pymc import mymodel S = pymc.MCMC(mymodel, db='pickle') S.sample(iter=10000, burn=5000, thin=2) pymc.Matplot.plot(S) This example will generate 10000 posterior samples, thinned by a factor of 2, with the first half discarded as burn-in. The sample is stored in a Python serialization (pickle) database. 4 Chapter 1. Introduction PyMC Documentation, Release 2.3.5 1.5 History PyMC began development in 2003, as an effort to generalize the process of building Metropolis-Hastings samplers, with an aim to making Markov chain Monte Carlo (MCMC) more accessible to non-statisticians (particularly ecologists). The choice to develop PyMC as a python module, rather than a standalone application, allowed the use MCMC methods in a larger modeling framework. By 2005, PyMC was reliable enough for version 1.0 to be released to the public. A small group of regular users, most associated with the University of Georgia, provided much of the feedback necessary for the refinement of PyMC to a usable state. In 2006, David Huard and Anand Patil joined Chris Fonnesbeck on the development team for PyMC 2.0. This iteration of the software strives for more flexibility, better performance and a better end-user experience than any previous version of PyMC. PyMC 2.2 was released in April 2012. It contains numerous bugfixes and optimizations, as well as a few new features, inculding improved output plotting, csv table output, improved imputation syntax, and posterior predictive check plots. PyMC 2.3 was released on October 31, 2013. It included Python 3 compatibility, the addition of the half-Cauchy distribution, improved summary plots, and some important bug fixes. This user guide has been updated for version 2.3. 1.6 Relationship to other packages PyMC in one of many general-purpose MCMC packages. The most prominent among them is WinBUGS, which has made MCMC (and with it, Bayesian statistics) accessible to a huge user community. Unlike PyMC, WinBUGS is a stand-alone, self-contained application. This can be an attractive feature for users without much programming experience, but others may find it constraining. A related package is JAGS, which provides a more UNIX-like implementation of the BUGS language. Other packages include Hierarchical Bayes Compiler and, more recently, STAN. It would be difficult to meaningfully benchmark PyMC against these other packages because of the unlimited variety in Bayesian probability models and flavors of the MCMC algorithm. However, it is possible to anticipate how it will perform in broad terms. PyMC’s number-crunching is done using a combination of industry-standard libraries (NumPy and the linear algebra libraries on which it depends) and hand-optimized Fortran routines. For models that are composed of variables valued as large arrays, PyMC will spend most of its time in these fast routines. In that case, it will be roughly as fast as packages written entirely in C and faster than WinBUGS. For finer-grained models containing mostly scalar variables, it will spend most of its time in coordinating Python code. In that case, despite our best efforts at optimization, PyMC will be significantly slower than packages written in C and on par with or slower than WinBUGS. However, as finegrained models are often small and simple, the total time required for sampling is often quite reasonable despite this poorer performance. We have chosen to spend time developing PyMC rather than using an existing package primarily because it allows us to build and efficiently fit any model we like within a productuve Python environment. We have emphasized extensibility throughout PyMC’s design, so if it doesn’t meet your needs out of the box, chances are you can make it do so with a relatively small amount of code. See the testimonials page on the wiki for reasons why other users have chosen PyMC. 1.7 Getting started This guide provides all the information needed to install PyMC, code a Bayesian statistical model, run the sampler, save and visualize the results. In addition, it contains a list of the statistical distributions currently available. More examples and tutorials are available from the PyMC web site. 1.5. History 5 PyMC Documentation, Release 2.3.5 6 Chapter 1. Introduction CHAPTER 2 Installation Date 1 July 2014 Authors Chris Fonnesbeck, Anand Patil, David Huard, John Salvatier Contact [email protected] Web site http://github.com/pymc-devs/pymc Copyright This document has been placed in the public domain. License PyMC is released under the Academic Free license. Version 2.3 PyMC is known to run on Mac OS X, Linux and Windows, but in theory should be able to work on just about any platform for which Python, a Fortran compiler and the NumPy module are available. However, installing some extra depencies can greatly improve PyMC’s performance and versatility. The following describes the required and optional dependencies and takes you through the installation process. 2.1 Dependencies PyMC requires some prerequisite packages to be present on the system. Fortunately, there are currently only a few hard dependencies, and all are freely available online. • Python version 2.6 or later. • NumPy (1.6 or newer): The fundamental scientific programming package, it provides a multidimensional array type and many useful functions for numerical analysis. • Matplotlib (1.0 or newer): 2D plotting library which produces publication quality figures in a variety of image formats and interactive environments • SciPy (optional): Library of algorithms for mathematics, science and engineering. • pyTables (optional): Package for managing hierarchical datasets and designed to efficiently and easily cope with extremely large amounts of data. Requires the HDF5 library. • pydot (optional): Python interface to Graphviz’s Dot language, it allows PyMC to create both directed and non-directed graphical representations of models. Requires the Graphviz library. • IPython (optional): An enhanced interactive Python shell and an architecture for interactive parallel computing. • nose (optional): A test discovery-based unittest extension (required to run the test suite). 7 PyMC Documentation, Release 2.3.5 There are prebuilt distributions that include all required dependencies. For Mac OS X and Windows users, we recommend the Anaconda Python distribution. Anaconda comes bundled with most of these prerequisites. Note that depending on the currency of these distributions, some packages may need to be updated manually. If, instead of installing the prebuilt binaries, you prefer (or have) to build pymc yourself, make sure you have a Fortran and a C compiler. There are free compilers (gfortran, gcc) available on all platforms. Other compilers have not been tested with PyMC but may work nonetheless. 2.2 Installation using EasyInstall The easiest way to install PyMC is to type in a terminal: easy_install pymc Provided EasyInstall (part of the setuptools module) is installed and in your path, this should fetch and install the package from the Python Package Index. Make sure you have the appropriate administrative privileges to install software on your computer. 2.3 Installing from pre-built binaries Pre-built binaries are available for Windows XP and Mac OS X. These can be installed as follows: 1. Download the installer for your platform from PyPI or the GitHub download page. 2. Double-click the executable installation package, then follow the on-screen instructions. For other platforms, you will need to build the package yourself from source. Fortunately, this should be relatively straightforward. 2.3.1 Anaconda If you are running the Anaconda Python distribution you can install a PyMC binary from the Binstar package management service, using the conda utility: conda install -c https://conda.binstar.org/pymc pymc 2.4 Compiling the source code First download the source code from GitHub and unpack it. Then move into the unpacked directory and follow the platform specific instructions. 2.4.1 Windows One way to compile PyMC on Windows is to install MinGW and MSYS. MinGW is the GNU Compiler Collection (GCC) augmented with Windows specific headers and libraries. MSYS is a POSIX-like console (bash) with UNIX command line tools. Download the Automated MinGW Installer and double-click on it to launch the installation process. You will be asked to select which components are to be installed: make sure the g77 compiler is selected and proceed with the instructions. Then download and install MSYS-1.0.exe, launch it and again follow the on-screen instructions. Once this is done, launch the MSYS console, change into the PyMC directory and type: 8 Chapter 2. Installation PyMC Documentation, Release 2.3.5 python setup.py install This will build the C and Fortran extension and copy the libraries and python modules in the site-packages directory of your Python distribution. Some Windows users have reported problems building PyMC with MinGW, particularly under Enthought Python. An alternative approach in this case is to use the gcc and gfortran compilers that are bundled with EPD (located in the Scripts directory). In order to do this, you should add the EPD “Scripts” directory to your PATH environment variable (ensuring that it appears ahead of the MinGW binary directory, if it exists on your PATH). Then build PyMC using the install command above. Alternatively, one may build the currently-available release of PyMC using pip. 2.4.2 Mac OS X or Linux In a terminal, type: python setup.py config_fc --fcompiler gfortran build python setup.py install The above syntax also assumes that you have gFortran installed and available. The sudo command may be required to install PyMC into the Python site-packages directory if it has restricted privileges. In addition, the python-dev package may be required to install PyMC on Linux systems. 2.5 Installing from GitHub You can check out the 2.3 branch of PyMC from the GitHub repository: git clone -b 2.3 git://github.com/pymc-devs/[email protected] Previous versions are available in the /tags directory. 2.6 Running the test suite pymc comes with a set of tests that verify that the critical components of the code work as expected. To run these tests, users must have nose installed. The tests are launched from a python shell: import pymc pymc.test() In case of failures, messages detailing the nature of these failures will appear. In case this happens (it shouldn’t), please report the problems on the issue tracker (the issues tab on the GitHub page), specifying the version you are using and the environment. 2.7 Bugs and feature requests Report problems with the installation, bugs in the code or feature request at the issue tracker. Comments and questions are welcome and should be addressed to PyMC’s mailing list. 2.5. Installing from GitHub 9 PyMC Documentation, Release 2.3.5 10 Chapter 2. Installation CHAPTER 3 Tutorial This tutorial will guide you through a typical PyMC application. Familiarity with Python is assumed, so if you are new to Python, books such as [Lutz2007] or [Langtangen2009] are the place to start. Plenty of online documentation can also be found on the Python documentation page. 3.1 An example statistical model Consider the following dataset, which is a time series of recorded coal mining disasters in the UK from 1851 to 1962 [Jarrett1979]. Fig. 3.1: Recorded coal mining disasters in the UK. Occurrences of disasters in the time series is thought to be derived from a Poisson process with a large rate parameter in the early part of the time series, and from one with a smaller rate in the later part. We are interested in locating the change point in the series, which perhaps is related to changes in mining safety regulations. 11 PyMC Documentation, Release 2.3.5 We represent our conceptual model formally as a statistical model: {︂ 𝑒 if 𝑡 < 𝑠 (𝐷𝑡 |𝑠, 𝑒, 𝑙) ∼ Poisson (𝑟𝑡 ) , 𝑟𝑡 = , 𝑙 if 𝑡 ≥ 𝑠 𝑠 ∼ Discrete Uniform(𝑡𝑙 , 𝑡ℎ ) 𝑒 ∼ Exponential(𝑟𝑒 ) 𝑙 ∼ Exponential(𝑟𝑙 ) 𝑡 ∈ [𝑡𝑙 , 𝑡ℎ ] The symbols are defined as: • 𝐷𝑡 : The number of disasters in year 𝑡. • 𝑟𝑡 : The rate parameter of the Poisson distribution of disasters in year 𝑡. • 𝑠: The year in which the rate parameter changes (the switchpoint). • 𝑒: The rate parameter before the switchpoint 𝑠. • 𝑙: The rate parameter after the switchpoint 𝑠. • 𝑡𝑙 , 𝑡ℎ : The lower and upper boundaries of year 𝑡. • 𝑟𝑒 , 𝑟𝑙 : The rate parameters of the priors of the early and late rates, respectively. Because we have defined 𝐷 by its dependence on 𝑠, 𝑒 and 𝑙, the latter three are known as the “parents” of 𝐷 and 𝐷 is called their “child”. Similarly, the parents of 𝑠 are 𝑡𝑙 and 𝑡ℎ , and 𝑠 is the child of 𝑡𝑙 and 𝑡ℎ . 3.2 Two types of variables At the model-specification stage (before the data are observed), 𝐷, 𝑠, 𝑒, 𝑟 and 𝑙 are all random variables. Bayesian “random” variables have not necessarily arisen from a physical random process. The Bayesian interpretation of probability is epistemic, meaning random variable 𝑥‘s probability distribution 𝑝(𝑥) represents our knowledge and uncertainty about 𝑥‘s value [Jaynes2003]. Candidate values of 𝑥 for which 𝑝(𝑥) is high are relatively more probable, given what we know. Random variables are represented in PyMC by the classes Stochastic and Deterministic. The only Deterministic in the model is 𝑟. If we knew the values of 𝑟‘s parents (𝑠, 𝑙 and 𝑒), we could compute the value of 𝑟 exactly. A Deterministic like 𝑟 is defined by a mathematical function that returns its value given values for its parents. Deterministic variables are sometimes called the systemic part of the model. The nomenclature is a bit confusing, because these objects usually represent random variables; since the parents of 𝑟 are random, 𝑟 is random also. A more descriptive (though more awkward) name for this class would be DeterminedByValuesOfParents. On the other hand, even if the values of the parents of variables switchpoint, disasters (before observing the data), early_mean or late_mean were known, we would still be uncertain of their values. These variables are characterized by probability distributions that express how plausible their candidate values are, given values for their parents. The Stochastic class represents these variables. A more descriptive name for these objects might be RandomEvenGivenValuesOfParents. We can represent model (??) in a file called disaster_model.py (the actual file can be found in pymc/examples/) as follows. First, we import the PyMC and NumPy namespaces: from pymc import DiscreteUniform, Exponential, deterministic, Poisson, Uniform import numpy as np Notice that from pymc we have only imported a select few objects that are needed for this particular model, whereas the entire numpy namespace has been imported, and conveniently given a shorter name. Objects from NumPy are subsequently accessed by prefixing np. to the name. Either approach is acceptable. Next, we enter the actual data values into an array: 12 Chapter 3. Tutorial PyMC Documentation, Release 2.3.5 disasters_array = \ np.array([ 4, 5, 3, 2, 1, 0, 3, 0, 4, 3, 2, 0, 1, 3, 0, 0, 5, 3, 1, 0, 1, 0, 1, 4, 4, 1, 1, 1, 1, 4, 5, 2, 0, 0, 2, 0, 3, 3, 1, 0, 0, 1, 0, 4, 1, 3, 3, 0, 1, 0, 0, 4, 2, 1, 2, 1, 0, 6, 4, 2, 0, 1, 1, 0, 3, 1, 1, 3, 0, 2, 1, 3, 5, 1, 2, 0, 4, 0, 4, 5, 1, 2, 0, 2, 0, 0, 3, 1, 0, 1, 0, 1, 2, 4, 3, 1, 1, 0, 0, 6, 2, 5, 0, 0, 1, 1, 0, 2, 1, 4, 1]) Note that you don’t have to type in this entire array to follow along; the code is available in the source tree, in this example script. Next, we create the switchpoint variable switchpoint switchpoint = DiscreteUniform('switchpoint', lower=0, upper=110, doc='Switchpoint[year]') DiscreteUniform is a subclass of Stochastic that represents uniformly-distributed discrete variables. Use of this distribution suggests that we have no preference a priori regarding the location of the switchpoint; all values are equally likely. Now we create the exponentially-distributed variables early_mean and late_mean for the early and late Poisson rates, respectively: early_mean = Exponential('early_mean', beta=1.) late_mean = Exponential('late_mean', beta=1.) Next, we define the variable rate, which selects the early rate early_mean for times before switchpoint and the late rate late_mean for times after switchpoint. We create rate using the deterministic decorator, which converts the ordinary Python function rate into a Deterministic object.: @deterministic(plot=False) def rate(s=switchpoint, e=early_mean, l=late_mean): ''' Concatenate Poisson means ''' out = np.empty(len(disasters_array)) out[:s] = e out[s:] = l return out The last step is to define the number of disasters disasters. This is a stochastic variable but unlike switchpoint, early_mean and late_mean we have observed its value. To express this, we set the argument observed to True (it is set to False by default). This tells PyMC that this object’s value should not be changed: disasters = Poisson('disasters', mu=rate, value=disasters_array, observed=True) 3.2.1 Why are data and unknown variables represented by the same object? Since its represented by a Stochastic object, disasters is defined by its dependence on its parent rate even though its value is fixed. This isn’t just a quirk of PyMC’s syntax; Bayesian hierarchical notation itself makes no distinction between random variables and data. The reason is simple: to use Bayes’ theorem to compute the posterior 𝑝(𝑒, 𝑠, 𝑙 | 𝐷) of model (??), we require the likelihood 𝑝(𝐷 | 𝑒, 𝑠, 𝑙). Even though disasters‘s value is known and fixed, we need to formally assign it a probability distribution as if it were a random variable. Remember, the likelihood and the probability function are essentially the same, except that the former is regarded as a function of the parameters and the latter as a function of the data. This point can be counterintuitive at first, as many peoples’ instinct is to regard data as fixed a priori and unknown variables as dependent on the data. One way to understand this is to think of statistical models like (??) as predictive models for data, or as models of the processes that gave rise to data. Before observing the value of disasters, we could have sampled from its prior predictive distribution 𝑝(𝐷) (i.e. the marginal distribution of the data) as follows: • Sample early_mean, switchpoint and late_mean from their priors. • Sample disasters conditional on these values. 3.2. Two types of variables 13 PyMC Documentation, Release 2.3.5 Even after we observe the value of disasters, we need to use this process model to make inferences about early_mean , switchpoint and late_mean because it’s the only information we have about how the variables are related. 3.3 Parents and children We have above created a PyMC probability model, which is simply a linked collection of variables. To see the nature of the links, import or run disaster_model.py and examine switchpoint‘s parents attribute from the Python prompt: >>> from pymc.examples import disaster_model >>> disaster_model.switchpoint.parents {'lower': 0, 'upper': 110} The parents dictionary shows us the distributional parameters of switchpoint, which are constants. Now let’s examine disasters‘s parents: >>> disaster_model.disasters.parents {'mu': <pymc.PyMCObjects.Deterministic 'rate' at 0x10623da50>} We are using rate as a distributional parameter of disasters (i.e. rate is disasters‘s parent). disasters internally labels rate as mu, meaning rate plays the role of the rate parameter in disasters‘s Poisson distribution. Now examine rate‘s children attribute: >>> disaster_model.rate.children set([<pymc.distributions.Poisson 'disasters' at 0x10623da90>]) Because disasters considers rate its parent, rate considers disasters its child. Unlike parents, children is a set (an unordered collection of objects); variables do not associate their children with any particular distributional role. Try examining the parents and children attributes of the other parameters in the model. The following directed acyclic graph is a visualization of the parent-child relationships in the model. Unobserved stochastic variables switchpoint, early_mean and late_mean are open ellipses, observed stochastic variable disasters is a filled ellipse and deterministic variable rate is a triangle. Arrows point from parent to child and display the label that the child assigns to the parent. See section Graphing models for more details. As the examples above have shown, pymc objects need to have a name assigned, such as switchpoint, early_mean or late_mean. These names are used for storage and post-processing: • as keys in on-disk databases, • as node labels in model graphs, • as axis labels in plots of traces, • as table labels in summary statistics. A model instantiated with variables having identical names raises an error to avoid name conflicts in the database storing the traces. In general however, pymc uses references to the objects themselves, not their names, to identify variables. 3.4 Variables’ values and log-probabilities All PyMC variables have an attribute called value that stores the current value of that variable. Try examining disasters‘s value, and you’ll see the initial value we provided for it: 14 Chapter 3. Tutorial PyMC Documentation, Release 2.3.5 switchpoint early_mean e s late_mean l rate mu disasters Fig. 3.2: Directed acyclic graph of the relationships in the coal mining disaster model example. >>> disaster_model.disasters.value array([4, 5, 4, 0, 1, 4, 3, 4, 0, 6, 4, 4, 1, 5, 5, 3, 4, 2, 5, 2, 0, 0, 1, 0, 1, 1, 0, 0, 3, 1, 0, 0, 2, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 4, 0, 0, 0, 1, 0, 0, 3, 2, 0, 2, 0, 3, 3, 3, 3, 0, 4, 4, 2, 3, 0, 0, 2, 2, 1, 1, 2, 1, 0, 1, 0, 6, 3, 1, 2, 0, 3, 2, 1, 1, 1, 3, 2, 1, 1, 0, 5, 4, 1, 1, 0, 1, 1, 1, 1]) 5, 1, 0, 2, 3, 1, 1, 4, 1, 3, 0, 2, If you check the values of early_mean, switchpoint and late_mean, you’ll see random initial values generated by PyMC: >>> disaster_model.switchpoint.value 44 >>> disaster_model.early_mean.value 0.33464706250079584 >>> disaster_model.late_mean.value 2.6491936762267811 Of course, since these are Stochastic elements, your values will be different than these. If you check rate‘s value, you’ll see an array whose first switchpoint elements are early_mean (here 0.33464706), and whose remaining elements are late_mean (here 2.64919368): >>> disaster_model.rate.value array([ 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 3.4. Variables’ values and log-probabilities 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 0.33464706, 2.64919368, 15 PyMC Documentation, Release 2.3.5 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368, 2.64919368]) To compute its value, rate calls the function we used to create it, passing in the values of its parents. Stochastic objects can evaluate their probability mass or density functions at their current values given the values of their parents. The logarithm of a stochastic object’s probability mass or density can be accessed via the logp attribute. For vector-valued variables like disasters, the logp attribute returns the sum of the logarithms of the joint probability or density of all elements of the value. Try examining switchpoint‘s and disasters‘s log-probabilities and early_mean ‘s and late_mean‘s log-densities: >>> disaster_model.switchpoint.logp -4.7095302013123339 >>> disaster_model.disasters.logp -1080.5149888046033 >>> disaster_model.early_mean.logp -0.33464706250079584 >>> disaster_model.late_mean.logp -2.6491936762267811 Stochastic objects need to call an internal function to compute their logp attributes, as rate needed to call an internal function to compute its value. Just as we created rate by decorating a function that computes its value, it’s possible to create custom Stochastic objects by decorating functions that compute their log-probabilities or densities (see chapter Building models). Users are thus not limited to the set of statistical distributions provided by PyMC. 3.4.1 Using Variables as parents of other Variables Let’s take a closer look at our definition of rate: @deterministic(plot=False) def rate(s=switchpoint, e=early_mean, l=late_mean): ''' Concatenate Poisson means ''' out = np.empty(len(disasters_array)) out[:s] = e out[s:] = l return out The arguments switchpoint, early_mean and late_mean are Stochastic objects, not numbers. If that is so, why aren’t errors raised when we attempt to slice array out up to a Stochastic object? Whenever a variable is used as a parent for a child variable, PyMC replaces it with its value attribute when the child’s value or log-probability is computed. When rate‘s value is recomputed, s.value is passed to the function as argument switchpoint. To see the values of the parents of rate all together, look at rate.parents.value. 16 Chapter 3. Tutorial PyMC Documentation, Release 2.3.5 3.5 Fitting the model with MCMC PyMC provides several objects that fit probability models (linked collections of variables) like ours. The primary such object, MCMC, fits models with a Markov chain Monte Carlo algorithm [Gamerman1997]. To create an MCMC object to handle our model, import disaster_model.py and use it as an argument for MCMC: >>> from pymc.examples import disaster_model >>> from pymc import MCMC >>> M = MCMC(disaster_model) In this case M will expose variables switchpoint, early_mean, late_mean and disasters as attributes; that is, M.switchpoint will be the same object as disaster_model.switchpoint. To run the sampler, call the MCMC object’s sample() (or isample(), for interactive sampling) method with arguments for the number of iterations, burn-in length, and thinning interval (if desired): >>> M.sample(iter=10000, burn=1000, thin=10) After a few seconds, you should see that sampling has finished normally. The model has been fitted. 3.5.1 What does it mean to fit a model? Fitting a model means characterizing its posterior distribution somehow. In this case, we are trying to represent the posterior 𝑝(𝑠, 𝑒, 𝑙|𝐷) by a set of joint samples from it. To produce these samples, the MCMC sampler randomly updates the values of switchpoint, early_mean and late_mean according to the Metropolis-Hastings algorithm [Gelman2004] over a specified number of iterations (iter). As the number of samples grows sufficiently large, the MCMC distributions of switchpoint, early_mean and late_mean converge to their joint stationary distribution. In other words, their values can be considered as random draws from the posterior 𝑝(𝑠, 𝑒, 𝑙|𝐷). PyMC assumes that the burn parameter specifies a sufficiently large number of iterations for the algorithm to converge, so it is up to the user to verify that this is the case (see chapter Model checking and diagnostics). Consecutive values sampled from switchpoint, early_mean and late_mean are always serially dependent, since it is a Markov chain. MCMC often results in strong autocorrelation among samples that can result in imprecise posterior inference. To circumvent this, it is useful to thin the sample by only retaining every k th sample, where 𝑘 is an integer value. This thinning interval is passed to the sampler via the thin argument. If you are not sure ahead of time what values to choose for the burn and thin parameters, you may want to retain all the MCMC samples, that is to set burn=0 and thin=1, and then discard the burn-in period and thin the samples after examining the traces (the series of samples). See [Gelman2004] for general guidance. 3.5.2 Accessing the samples The output of the MCMC algorithm is a trace, the sequence of retained samples for each variable in the model. These traces can be accessed using the trace(name, chain=-1) method. For example: >>> M.trace('switchpoint')[:] array([41, 40, 40, ..., 43, 44, 44]) The trace slice [start:stop:step] works just like the NumPy array slice. By default, the returned trace array contains the samples from the last call to sample, that is, chain=-1, but the trace from previous sampling runs can be retrieved by specifying the correspondent chain index. To return the trace from all chains, simply use chain=None. 1 1 Note that the unknown variables switchpoint, early_mean, 3.5. Fitting the model with MCMC 17 PyMC Documentation, Release 2.3.5 3.5.3 Sampling output You can examine the marginal posterior of any variable by plotting a histogram of its trace: >>> from pylab import hist, show >>> hist(M.trace('late_mean')[:]) (array([ 8, 52, 565, 1624, 2563, 2105, 1292, 488, 258, 45]), array([ 0.52721865, 0.60788251, 0.68854637, 0.76921023, 0.84987409, 0.93053795, 1.01120181, 1.09186567, 1.17252953, 1.25319339]), <a list of 10 Patch objects>) >>> show() You should see something like this: Fig. 3.3: Histogram of the marginal posterior probability of parameter late_mean. PyMC has its own plotting functionality, via the optional matplotlib module as noted in the installation notes. The Matplot module includes a plot function that takes the model (or a single parameter) as an argument: >>> from pymc.Matplot import plot >>> plot(M) For each variable in the model, plot generates a composite figure, such as this one for the switchpoint in the disasters model: The upper left-hand pane of this figure shows the temporal series of the samples from switchpoint, while below 18 Chapter 3. Tutorial PyMC Documentation, Release 2.3.5 Fig. 3.4: Temporal series, autocorrelation plot and histogram of the samples drawn for switchpoint. is an autocorrelation plot of the samples. The right-hand pane shows a histogram of the trace. The trace is useful for evaluating and diagnosing the algorithm’s performance (see [Gelman1996]), while the histogram is useful for visualizing the posterior. For a non-graphical summary of the posterior, simply call M.stats(). 3.5.4 Imputation of Missing Data As with most textbook examples, the models we have examined so far assume that the associated data are complete. That is, there are no missing values corresponding to any observations in the dataset. However, many real-world datasets have missing observations, usually due to some logistical problem during the data collection process. The easiest way of dealing with observations that contain missing values is simply to exclude them from the analysis. However, this results in loss of information if an excluded observation contains valid values for other quantities, and can bias results. An alternative is to impute the missing values, based on information in the rest of the model. For example, consider a survey dataset for some wildlife species: Count 15 10 6 Site 1 1 1 Observer 1 2 1 Temperature 15 NA 11 Each row contains the number of individuals seen during the survey, along with three covariates: the site on which the survey was conducted, the observer that collected the data, and the temperature during the survey. If we are interested in modelling, say, population size as a function of the count and the associated covariates, it is difficult to accommodate the second observation because the temperature is missing (perhaps the thermometer was broken that day). Ignoring this observation will allow us to fit the model, but it wastes information that is contained in the other covariates. 3.5. Fitting the model with MCMC 19 PyMC Documentation, Release 2.3.5 In a Bayesian modelling framework, missing data are accommodated simply by treating them as unknown model parameters. Values for the missing data 𝑦˜ are estimated naturally, using the posterior predictive distribution: ∫︁ 𝑝(˜ 𝑦 |𝑦) = 𝑝(˜ 𝑦 |𝜃)𝑓 (𝜃|𝑦)𝑑𝜃 This describes additional data 𝑦˜, which may either be considered unobserved data or potential future observations. We can use the posterior predictive distribution to model the likely values of missing data. Consider the coal mining disasters data introduced previously. Assume that two years of data are missing from the time series; we indicate this in the data array by the use of an arbitrary placeholder value, None.: x = np.array([ 3, 3, 5, 4, 5, 2, 2, 3, 4, 2, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 3, 3, 1, None, 0, 0, 0, 1, 0, 4, 3, 1, 0, 0, 2, 0, 5, 1, 3, 3, 0, 1, 0, 4, 0, 4, 4, None, 1, 0, 2, 1, 1, 1, 0, 0, 1, 1, 2, 3, 0, 1, 1, 4, 5, 1, 2, 0, 2, 0, 3, 5, 1, 2, 0, 4, 0, 4, 3, 1, 0, 1, 2, 1, 0, 4, 1, 1, 1, 0, 0, 6, 3, 3, 4, 0, 2, 6, 2, 5, 3, 0, 0, 1, 1, 0, 2, 0, 1, 4, 1]) To estimate these values in PyMC, we generate a masked array. These are specialised NumPy arrays that contain a matching True or False value for each element to indicate if that value should be excluded from any computation. Masked arrays can be generated using NumPy’s ma.masked_equal function: >>> masked_values = np.ma.masked_equal(x, value=None) >>> masked_values masked_array(data = [4 5 4 0 1 4 3 4 0 6 3 3 4 0 2 6 3 3 5 4 5 3 1 4 4 1 5 5 3 4 2 5 2 2 3 4 2 1 3 -- 2 1 1 1 1 3 0 0 1 0 1 1 0 0 3 1 0 3 2 2 0 1 1 1 0 1 0 1 0 0 0 2 1 0 0 0 1 1 0 2 3 3 1 -- 2 1 1 1 1 2 4 2 0 0 1 4 0 0 0 1 0 0 0 0 0 1 0 0 1 0 1], mask = [False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False True False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False False True False False False False False False False False False False False False False False False False False False False False False False False False False False False], fill_value=?) This masked array, in turn, can then be passed to one of PyMC’s data stochastic variables, which recognizes the masked array and replaces the missing values with Stochastic variables of the desired type. For the coal mining disasters problem, recall that disaster events were modeled as Poisson variates: >>> from pymc import Poisson >>> disasters = Poisson('disasters', mu=rate, value=masked_values, observed=True) Here rate is an array of means for each year of data, allocated according to the location of the switchpoint. Each element in disasters is a Poisson Stochastic, irrespective of whether the observation was missing or not. The difference is that actual observations are data Stochastics (observed=True), while the missing values are non-data Stochastics. The latter are considered unknown, rather than fixed, and therefore estimated by the MCMC algorithm, just as unknown model parameters. The entire model looks very similar to the original model: # Switchpoint switch = DiscreteUniform('switch', lower=0, upper=110) # Early mean 20 Chapter 3. Tutorial PyMC Documentation, Release 2.3.5 early_mean = Exponential('early_mean', beta=1) # Late mean late_mean = Exponential('late_mean', beta=1) @deterministic(plot=False) def rate(s=switch, e=early_mean, l=late_mean): """Allocate appropriate mean to time series""" out = np.empty(len(disasters_array)) # Early mean prior to switchpoint out[:s] = e # Late mean following switchpoint out[s:] = l return out # The inefficient way, using the Impute function: # D = Impute('D', Poisson, disasters_array, mu=r) # # The efficient way, using masked arrays: # Generate masked array. Where the mask is true, # the value is taken as missing. masked_values = masked_array(disasters_array, mask=disasters_array==-999) # Pass masked array to data stochastic, and it does the right thing disasters = Poisson('disasters', mu=rate, value=masked_values, observed=True) Here, we have used the masked_array function, rather than masked_equal, and the value -999 as a placeholder for missing data. The result is the same. Fig. 3.5: Trace, autocorrelation plot and posterior distribution of the missing data points in the example. 3.5. Fitting the model with MCMC 21 PyMC Documentation, Release 2.3.5 3.6 Fine-tuning the MCMC algorithm MCMC objects handle individual variables via step methods, which determine how parameters are updated at each step of the MCMC algorithm. By default, step methods are automatically assigned to variables by PyMC. To see which step methods 𝑀 is using, look at its step_method_dict attribute with respect to each parameter: >>> M.step_method_dict[disaster_model.switchpoint] [<pymc.StepMethods.DiscreteMetropolis object at 0x3e8cb50>] >>> M.step_method_dict[disaster_model.early_mean] [<pymc.StepMethods.Metropolis object at 0x3e8cbb0>] >>> M.step_method_dict[disaster_model.late_mean] [<pymc.StepMethods.Metropolis object at 0x3e8ccb0>] The value of step_method_dict corresponding to a particular variable is a list of the step methods 𝑀 is using to handle that variable. You can force 𝑀 to use a particular step method by calling M.use_step_method before telling it to sample. The following call will cause 𝑀 to handle late_mean with a standard Metropolis step method, but with proposal standard deviation equal to 2: >>> from pymc import Metropolis >>> M.use_step_method(Metropolis, disaster_model.late_mean, proposal_sd=2.) Another step method class, AdaptiveMetropolis, is better at handling highly-correlated variables. If your model mixes poorly, using AdaptiveMetropolis is a sensible first thing to try. 3.7 Beyond the basics That was a brief introduction to basic PyMC usage. Many more topics are covered in the subsequent sections, including: • Class Potential, another building block for probability models in addition to Stochastic and Deterministic • Normal approximations • Using custom probability distributions • Object architecture • Saving traces to the disk, or streaming them to the disk during sampling • Writing your own step methods and fitting algorithms. Also, be sure to check out the documentation for the Gaussian process extension, which is available on PyMC’s download page. late_mean and rate will all accrue samples, but disasters will not because its value has been observed and is not updated. Hence disasters has no trace and calling M.trace(’disasters’)[:] will raise an error. 22 Chapter 3. Tutorial CHAPTER 4 Building models Bayesian inference begins with specification of a probability model relating unknown variables to data. PyMC provides three basic building blocks for Bayesian probability models: Stochastic, Deterministic and Potential. A Stochastic object represents a variable whose value is not completely determined by its parents, and a Deterministic object represents a variable that is entirely determined by its parents. In object-oriented programming parlance, Stochastic and Deterministic are subclasses of the Variable class, which only serves as a template for other classes and is never actually implemented in models. The third basic class, Potential, represents ‘factor potentials’ ([Lauritzen1990],[Jordan2004]_), which are not variables but simply log-likelihood terms and/or constraints that are multiplied into joint distributions to modify them. Potential and Variable are subclasses of Node. PyMC probability models are simply linked groups of Stochastic, Deterministic and Potential objects. These objects have very limited awareness of the models in which they are embedded and do not themselves possess methods for updating their values in fitting algorithms. Objects responsible for fitting probability models are described in chapter Fitting Models. 4.1 The Stochastic class A stochastic variable has the following primary attributes: value: The variable’s current value. logp: The log-probability of the variable’s current value given the values of its parents. A stochastic variable can optionally be endowed with a method called random, which draws a value for the variable given the values of its parents 1 . Stochastic objects have the following additional attributes: parents: A dictionary containing the variable’s parents. The keys of the dictionary correspond to the names assigned to the variable’s parents by the variable, and the values correspond to the actual parents. For example, the keys of 𝑠‘s parents dictionary in model ((??)) would be ’t_l’ and ’t_h’. Thanks to Python’s dynamic typing, the actual parents (i.e. the values of the dictionary) may be of any class or type. children: A set containing the variable’s children. extended_parents: A set containing all the stochastic variables on which the variable depends either directly or via a sequence of deterministic variables. If the value of any of these variables changes, the variable will need to recompute its log-probability. 1 Note that the random method does not provide a Gibbs sample unless 23 PyMC Documentation, Release 2.3.5 extended_children: A set containing all the stochastic variables and potentials that depend on the variable either directly or via a sequence of deterministic variables. If the variable’s value changes, all of these variables will need to recompute their log-probabilities. observed: A flag (boolean) indicating whether the variable’s value has been observed (is fixed). dtype: A NumPy dtype object (such as numpy.int) that specifies the type of the variable’s value to fitting methods. If this is None (default) then no type is enforced. 4.1.1 Creation of stochastic variables There are three main ways to create stochastic variables, called the automatic, decorator, and direct interfaces. Automatic Stochastic variables with standard distributions provided by PyMC (see chapter Probability distributions) can be created in a single line using special subclasses of Stochastic. For example, the uniformly-distributed discrete variable switchpoint in (disaster_model) is created using the automatic interface as follows: switchpoint = DiscreteUniform('switchpoint', lower=0, upper=110, doc='Switchpoint[year]') In addition to the classes in chapter Probability distributions, scipy.stats.distributions‘ random variable classes are wrapped as Stochastic subclasses if SciPy is installed. These distributions are in the submodule pymc.SciPyDistributions. Users can call the class factory stochastic_from_dist to produce Stochastic subclasses of their own from probability distributions not included with PyMC. Decorator Uniformly-distributed discrete stochastic variable switchpoint in (disaster_model) could alternatively be created from a function that computes its log-probability as follows: @pymc.stochastic(dtype=int) def switchpoint(value=1900, t_l=1851, t_h=1962): """The switchpoint for the rate of disaster occurrence.""" if value > t_h or value < t_l: # Invalid values return -np.inf else: # Uniform log-likelihood return -np.log(t_h - t_l + 1) Note that this is a simple Python function preceded by a Python expression called a decorator [vanRossum2010], here called @stochastic. Generally, decorators enhance functions with additional properties or functionality. The Stochastic object produced by the @stochastic decorator will evaluate its log-probability using the function switchpoint. The value argument, which is required, provides an initial value for the variable. The remaining arguments will be assigned as parents of switchpoint (i.e. they will populate the parents dictionary). To emphasize, the Python function decorated by @stochastic should compute the log-density or logprobability of the variable. That is why the return value in the example above is − log(𝑡ℎ − 𝑡𝑙 + 1) rather than 1/(𝑡ℎ − 𝑡𝑙 + 1). The value and parents of stochastic variables may be any objects, provided the log-probability function returns a real number (float). PyMC and SciPy both provide implementations of several standard probability distributions that may be helpful for creating custom stochastic variables. The decorator stochastic can take any of the arguments Stochastic.__init__ takes except parents, logp, random, doc and value. These arguments include trace, plot, verbose, dtype, rseed and name. The decorator interface has a slightly more complex implementation which allows you to specify a random method for sampling the stochastic variable’s value conditional on its parents. 24 Chapter 4. Building models PyMC Documentation, Release 2.3.5 @pymc.stochastic(dtype=int) def switchpoint(value=1900, t_l=1851, t_h=1962): """The switchpoint for the rate of disaster occurrence.""" def logp(value, t_l, t_h): if value > t_h or value < t_l: return -np.inf else: return -np.log(t_h - t_l + 1) def random(t_l, t_h): from numpy.random import random return np.round( (t_l - t_h) * random() ) + t_l The stochastic variable again gets its name, docstring and parents from function switchpoint, but in this case it will evaluate its log-probability using the logp function. The random function will be used when switchpoint.random() is called. Note that random doesn’t take a value argument, as it generates values itself. Direct It’s possible to instantiate Stochastic directly: def switchpoint_logp(value, t_l, t_h): if value > t_h or value < t_l: return -np.inf else: return -np.log(t_h - t_l + 1) def switchpoint_rand(t_l, t_h): from numpy.random import random return np.round( (t_l - t_h) * random() ) + t_l switchpoint = Stochastic( logp = switchpoint_logp, doc = 'The switchpoint for the rate of disaster occurrence.', name = 'switchpoint', parents = {'t_l': 1851, 't_h': 1962}, random = switchpoint_rand, trace = True, value = 1900, dtype=int, rseed = 1., observed = False, cache_depth = 2, plot=True, verbose = 0) Notice that the log-probability and random variate functions are specified externally and passed to Stochastic as arguments. This is a rather awkward way to instantiate a stochastic variable; consequently, such implementations should be rare. 4.1. The Stochastic class 25 PyMC Documentation, Release 2.3.5 A Warning: Don’t update stochastic variables’ values in-place Stochastic objects’ values should not be updated in-place. This confuses PyMC’s caching scheme and corrupts the process used for accepting or rejecting proposed values in the MCMC algorithm. The only way a stochastic variable’s value should be updated is using statements of the following form: A.value = new_value The following are in-place updates and should _never_ be used: * ``A.value += 3`` * ``A.value[2,1] = 5`` * ``A.value.attribute = new_attribute_value`` This restriction becomes onerous if a step method proposes values for the elements of an array-valued variable separately. In this case, it may be preferable to partition the variable into several scalar-valued variables stored in an array or list. 4.2 Data Data are represented by Stochastic objects whose observed attribute is set to True. If a stochastic variable’s observed flag is True, its value cannot be changed, and it won’t be sampled by the fitting method. 4.2.1 Declaring stochastic variables to be data In each interface, an optional keyword argument observed can be set to True. In the decorator interface, this argument is added to the @stochastic decorator: @pymc.stochastic(observed=True) In the other interfaces, the observed=True argument is added to the instantiation of the Stochastic, or its subclass: x = pymc.Binomial('x', n=n, p=p, observed=True) Alternatively, in the decorator interface only, a Stochastic object’s observed flag can be set to true by stacking an @observed decorator on top of the @stochastic decorator: @pymc.observed(dtype=int) def ... 4.3 The Deterministic class The Deterministic class represents variables whose values are completely determined by the values of their parents. For example, in model (disaster_model), rate is a deterministic variable. Recall it was defined by {︂ 𝑒 𝑡≤𝑠 𝑟𝑡 = , 𝑙 𝑡>𝑠 so rate‘s value can be computed exactly from the values of its parents early_mean, late_mean and switchpoint. A deterministic variable’s most important attribute is value, which gives the current value of the variable given the values of its parents. Like Stochastic‘s logp attribute, this attribute is computed on-demand and cached for efficiency. 26 Chapter 4. Building models PyMC Documentation, Release 2.3.5 A Deterministic variable has the following additional attributes: parents: A dictionary containing the variable’s parents. The keys of the dictionary correspond to the names assigned to the variable’s parents by the variable, and the values correspond to the actual parents. children: A set containing the variable’s children, which must be nodes. Deterministic variables have no methods. 4.3.1 Creation of deterministic variables Deterministic variables are less complicated than stochastic variables, and have similar automatic, decorator, and direct interfaces: Automatic A handful of common functions have been wrapped in Deterministic objects. These are brief enough to list: LinearCombination: Has two parents 𝑥 and 𝑦, both of which must be iterable (i.e. vector-valued). This function returns: ∑︁ 𝑥𝑇𝑖 𝑦𝑖 . 𝑖 Index: Has two parents 𝑥 and index. 𝑥 must be iterables, index must be valued as an integer. The value of an index is: 𝑥[index]𝑇 𝑦[index]. Index is useful for implementing dynamic models, in which the parent-child connections change. Lambda: Converts an anonymous function (in Python, called lambda functions) to a Deterministic instance on a single line. CompletedDirichlet: PyMC represents Dirichlet variables of length 𝑘 by the first 𝑘 − 1 elements; since they must sum to 1, the 𝑘 𝑡ℎ element is determined by the others. CompletedDirichlet appends the 𝑘 𝑡ℎ element to the value of its parent 𝐷. Logit, InvLogit, StukelLogit, StukelInvLogit: Common link functions for generalized linear models, and their inverses. It’s a good idea to use these classes when feasible in order to give hints to step methods. Elementary operations on variables Certain elementary operations on variables create deterministic variables. For example: >>> x = pymc.MvNormalCov('x',np.ones(3),np.eye(3)) >>> y = pymc.MvNormalCov('y',np.ones(3),np.eye(3)) >>> print x+y <pymc.PyMCObjects.Deterministic '(x_add_y)' at 0x105c3bd10> >>> print x[0] <pymc.CommonDeterministics.Index 'x[0]' at 0x105c52390> >>> print x[1]+y[2] <pymc.PyMCObjects.Deterministic '(x[1]_add_y[2])' at 0x105c52410> All the objects thus created have trace=False and plot=False by default. This convenient method of generating simple deterministics was inspired by [Kerman2004]. 4.3. The Deterministic class 27 PyMC Documentation, Release 2.3.5 Decorator A deterministic variable can be created via a decorator in a way very similar to Stochastic‘s decorator interface: @deterministic(plot=False) def rate(s=switchpoint, e=early_mean, l=late_mean): ''' Concatenate Poisson means ''' out = empty(len(disasters_array)) out[:s] = e out[s:] = l return out Notice that rather than returning the log-probability, as is the case for Stochastic objects, the function returns the value of the deterministic object, given its parents. This return value may be of any type, as is suitable for the problem at hand. Also notice that, unlike for Stochastic objects, there is no value argument passed, since the value is calculated deterministically by the function itself. Arguments’ keys and values are converted into a parent dictionary as with Stochastic‘s short interface. The deterministic decorator can take trace, verbose and plot arguments, like the stochastic decorator 2 . Direct Deterministic objects can also be instantiated directly: def rate_eval(switchpoint = s, early_rate = e, late_rate = l): value = zeros(N) value[:switchpoint] = early_rate value[switchpoint:] = late_rate return value rate = pymc.Deterministic(eval = rate_eval, name = 'rate', parents = {'switchpoint': switchpoint, 'early_mean': early_mean, 'late_mean': late_mean}), doc = 'The rate of disaster occurrence.', trace = True, verbose = 0, dtype=float, plot=False, cache_depth = 2) 4.4 Containers In some situations it would be inconvenient to assign a unique label to each parent of some variable. Consider 𝑦 in the following model: 𝑥0 ∼ 𝑁 (0, 𝜏𝑥 ) 𝑥𝑖+1 |𝑥𝑖 ∼ N(𝑥𝑖 , 𝜏𝑥 ) 𝑖 = 0, . . . , 𝑁 − 2 (︃𝑁 −1 )︃ ∑︁ 𝑦|𝑥 ∼ 𝑁 𝑥2𝑖 , 𝜏𝑦 𝑖=0 Here, 𝑦 depends on every element of the Markov chain 𝑥, but we wouldn’t want to manually enter 𝑁 parent labels ‘x_0’, ‘x_1’, etc. This situation can be handled naturally in PyMC: 2 28 Note that deterministic variables have no observed flag. If a Chapter 4. Building models PyMC Documentation, Release 2.3.5 N = 10 x_0 = pymc.Normal(`x_0', mu=0, tau=1) x = np.empty(N, dtype=object) x[0] = x_0 for i in range(1, N): x[i] = pymc.Normal(`x_%i' % i, mu=x[i-1], tau=1) @pymc.observed def y(value=1, mu=x, tau=100): return pymc.normal_like(value, np.sum(mu**2), tau) PyMC automatically wraps array 𝑥 in an appropriate Container class. The expression ’x_%i’ % i labels each Normal object in the container with the appropriate index 𝑖. For example, if i=1, the name of the corresponding element becomes ‘x_1’. Containers, like variables, have an attribute called value. This attribute returns a copy of the (possibly nested) iterable that was passed into the container function, but with each variable inside replaced with its corresponding value. Containers can be constructed from lists, tuples, dictionaries, Numpy arrays, modules, sets or any object with a __dict__ attribute. Variables and non-variables can be freely mixed in these containers, and different types of containers can be nested 3 . Containers attempt to behave like the objects they wrap. All containers are subclasses of ContainerBase. Containers have the following useful attributes in addition to value: • variables • stochastics • potentials • deterministics • data_stochastics • step_methods. Each of these attributes is a set containing all the objects of each type in a container, and within any containers in the container. 4.5 The Potential class The joint density corresponding to model ((??)) can be written as follows: 𝑝(𝐷, 𝑠, 𝑙, 𝑒) = 𝑝(𝐷|𝑠, 𝑙, 𝑒)𝑝(𝑠)𝑝(𝑙)𝑝(𝑒). Each factor in the joint distribution is a proper, normalized probability distribution for one of the variables conditional on its parents. Such factors are contributed by Stochastic objects. In some cases, it’s nice to be able to modify the joint density by incorporating terms that don’t correspond to probabilities of variables conditional on parents, for example: 𝑝(𝑥0 , 𝑥2 , . . . 𝑥𝑁 −1 ) ∝ 𝑁 −2 ∏︁ 𝜓𝑖 (𝑥𝑖 , 𝑥𝑖+1 ). 𝑖=0 3 Nodes whose parents are containers make private shallow copies of those 4.5. The Potential class 29 PyMC Documentation, Release 2.3.5 In other cases we may want to add probability terms to existing models. For example, suppose we want to constrain the difference between 𝑒 and 𝑙 in ((??)) to be less than 1, so that the joint density becomes 𝑝(𝐷, 𝑠, 𝑙, 𝑒) ∝ 𝑝(𝐷|𝑠, 𝑙, 𝑒)𝑝(𝑠)𝑝(𝑙)𝑝(𝑒)𝐼(|𝑒 − 𝑙| < 1). Its possible to express this constraint by adding variables to the model, or by grouping 𝑒 and 𝑙 to form a vector-valued variable, but it’s uncomfortable to do so. Arbitrary factors such as 𝜓 and the indicator function 𝐼(|𝑒 − 𝑙| < 1) are implemented by objects of class Potential (from [Lauritzen1990] and [Jordan2004], who call these terms ‘factor potentials’). Bayesian hierarchical notation (cf model ((??))) doesn’t accomodate these potentials. They are often used in cases where there is no natural dependence hierarchy, such as the first example above (which is known as a Markov random field). They are also useful for expressing ‘soft data’ [Christakos2002] as in the second example below. Potentials have one important attribute, logp, the log of their current probability or probability density value given the values of their parents. The only other additional attribute of interest is parents, a dictionary containing the potential’s parents. Potentials have no methods. They have no trace attribute, because they are not variables. They cannot serve as parents of variables (for the same reason), so they have no children attribute. 4.5.1 An example of soft data The role of potentials can be confusing, so we will provide another example: we have a dataset 𝑡 consisting of the days on which several marked animals were recaptured. We believe that the probability 𝑆 that an animal is not recaptured on any given day can be explained by a covariate vector 𝑥. We model this situation as follows: 𝑡𝑖 |𝑆𝑖 ∼ Geometric(𝑆𝑖 ), −1 𝑆𝑖 = logit (𝛽𝑥𝑖 ), 𝑖 = 1...𝑁 𝑖 = 1...𝑁 𝛽 ∼ N(𝜇𝛽 , 𝑉𝛽 ). Now suppose we have some knowledge of other related experiments and we have a good idea of what 𝑆 will be independent of the value of 𝛽. It’s not obvious how to work this ‘soft data’, because as we’ve written the model 𝑆 is completely determined by 𝛽. There are three options within the strict Bayesian hierarchical framework: • Express the soft data as an informative prior on 𝛽. • Incorporate the data from the previous experiments explicitly into the model. • Refactor the model so that 𝑆 is at the bottom of the hierarchy, and assign the prior directly. Factor potentials provide a convenient way to incorporate the soft data without the need for such major modifications. We can simply modify the joint distribution from 𝑝(𝑡|𝑆(𝑥, 𝛽))𝑝(𝛽) to 𝛾(𝑆)𝑝(𝑡|𝑆(𝑥, 𝛽))𝑝(𝛽), where the value of 𝛾 is large if 𝑆‘s value is plausible (based on our external information) and small otherwise. We do not know the normalizing constant for the new distribution, but we don’t need it to use most popular fitting algorithms. It’s a good idea to check the induced priors on 𝑆 and 𝛽 for sanity. This can be done in PyMC by fitting the model with the data 𝑡 removed. Its important to understand that 𝛾 is not a variable, so it does not have a value. That means, among other things, there will be no 𝛾 column in MCMC traces. 30 Chapter 4. Building models PyMC Documentation, Release 2.3.5 4.5.2 Creation of Potentials There are two ways to create potentials: Decorator A potential can be created via a decorator in a way very similar to Deterministic‘s decorator interface: @pymc.potential def psi_i(x_lo = x[i], x_hi = x[i+1]): """A pair potential""" return -(xlo - xhi)**2 The function supplied should return the potential’s current log-probability or log-density as a Numpy float. The potential decorator can take verbose and cache_depth arguments like the stochastic decorator. Direct The same potential could be created directly as follows: def psi_i_logp(x_lo = x[i], x_hi = x[i+1]): return -(xlo - xhi)**2 psi_i = pymc.Potential(logp = psi_i_logp, name = 'psi_i', parents = {'xlo': x[i], 'xhi': x[i+1]}, doc = 'A pair potential', verbose = 0, cache_depth = 2) 4.6 Graphing models The function dag (or graph) in pymc.graph draws graphical representations of Model (Chapter Fitting Models) instances using GraphViz via the Python package PyDot. See [Lauritzen1990] and [Jordan2004] for more discussion of useful information that can be read off of graphical models 4 . The symbol for stochastic variables is an ellipse. Parent-child relationships are indicated by arrows. These arrows point from parent to child and are labeled with the names assigned to the parents by the children. PyMC’s symbol for deterministic variables is a downward-pointing triangle. A graphical representation of model (??) is shown in Directed acyclic graph of the relationships in the coal mining disaster model example.. Note that 𝐷 is shaded because it is flagged as data. The symbol for factor potentials is a rectangle, as in the following model. Factor potentials are usually associated with undirected grahical models. In undirected representations, each parent of a potential is connected to every other parent by an undirected edge. The undirected representation of the model pictured above is much more compact: Directed or mixed graphical models can be represented in an undirected form by ‘moralizing’, which is done by the function pymc.graph.moral_graph. 4.7 Class LazyFunction and caching This section gives an overview of how PyMC computes log-probabilities. This is advanced information that is not required in order to use PyMC. 4 Note that these authors do not consider deterministic variables. 4.6. Graphing models 31 PyMC Documentation, Release 2.3.5 chi p d p p c p gamma p p p psi e p p p p p b zeta p p phi p a Fig. 4.1: Directed graphical model example. Factor potentials are represented by rectangles and stochastic variables by ellipses. 32 Chapter 4. Building models PyMC Documentation, Release 2.3.5 The logp attributes of stochastic variables and potentials and the value attributes of deterministic variables are wrappers for instances of class LazyFunction. Lazy functions are wrappers for ordinary Python functions. A lazy function L could be created from a function fun as follows: L = pymc.LazyFunction(fun, arguments) The argument arguments is a dictionary container; fun must accept keyword arguments only. When L‘s get() method is called, the return value is the same as the call fun(**arguments.value) Note that no arguments need to be passed to L.get; lazy functions memorize their arguments. Before calling fun, L will check the values of arguments.variables against an internal cache. This comparison is done by reference, not by value, and this is part of the reason why stochastic variables’ values cannot be updated inplace. If arguments.variables‘ values match a frame of the cache, the corresponding output value is returned and fun is not called. If a call to fun is needed, arguments.variables‘ values and the return value replace the oldest frame in the cache. The depth of the cache can be set using the optional init argument cache_depth, which defaults to 2. Caching is helpful in MCMC, because variables’ log-probabilities and values tend to be queried multiple times for the same parental value configuration. The default cache depth of 2 turns out to be most useful in Metropolis-Hastingstype algorithms involving proposed values that may be rejected. Lazy functions are implemented in C using Pyrex, a language for writing Python extensions. the variable has no children. deterministic variable’s value were known, its parents would be restricted to the inverse image of that value under the deterministic variable’s evaluation function. This usage would be extremely difficult to support in general, but it can be implemented for particular applications at the StepMethod level. containers. This is done for technical reasons rather than to protect users from accidental misuse. 4.7. Class LazyFunction and caching 33 PyMC Documentation, Release 2.3.5 34 Chapter 4. Building models CHAPTER 5 Fitting Models PyMC provides three objects that fit models: • MCMC, which coordinates Markov chain Monte Carlo algorithms. The actual work of updating stochastic variables conditional on the rest of the model is done by StepMethod objects, which are described in this chapter. • MAP, which computes maximum a posteriori estimates. • NormApprox, which computes the ‘normal approximation’ [Gelman2004]: the joint distribution of all stochastic variables in a model is approximated as normal using local information at the maximum a posteriori estimate. All three objects are subclasses of Model, which is PyMC’s base class for fitting methods. MCMC and NormApprox, both of which can produce samples from the posterior, are subclasses of Sampler, which is PyMC’s base class for Monte Carlo fitting methods. Sampler provides a generic sampling loop method and database support for storing large sets of joint samples. These base classes implement some basic methods that are inherited by the three implemented fitting methods, so they are documented at the end of this section. 5.1 Creating models The first argument to any fitting method’s __init__ method, including that of MCMC, is called input. The input argument can be just about anything; once you have defined the nodes that make up your model, you shouldn’t even have to think about how to wrap them in a Model instance. Some examples of model instantiation using nodes a, b and c follow: • M = Model(set([a,b,c])) • M = Model({‘a’: a, ‘d’: 𝑎, and M.d will be [b,c]. [b,c]}) In this case, 𝑀 will expose 𝑎 and 𝑑 as attributes: M.a will be • M = Model([[a,b],c]) • If file MyModule contains the definitions of a, b and c: import MyModule M = Model(MyModule) In this case, 𝑀 will expose 𝑎, 𝑏 and 𝑐 as attributes. • Using a ‘model factory’ function: def make_model(x): a = pymc.Exponential('a', beta=x, value=0.5) @pymc.deterministic 35 PyMC Documentation, Release 2.3.5 def b(a=a): return 100-a @pymc.stochastic def c(value=.5, a=a, b=b): return (value-a)**2/b return locals() M = pymc.Model(make_model(3)) In this case, 𝑀 will also expose 𝑎, 𝑏 and 𝑐 as attributes. 5.2 The Model class This class serves as a container for probability models and as a base class for the classes responsible for model fitting, such as MCMC. Model‘s init method takes the following arguments: input: Some collection of PyMC nodes defining a probability model. These may be stored in a list, set, tuple, dictionary, array, module, or any object with a __dict__ attribute. verbose (optional): An integer controlling the verbosity of the model’s output. Models’ useful methods are: draw_from_prior(): Sets all stochastic variables’ values to new random values, which would be a sample from the joint distribution if all data and Potential instances’ log-probability functions returned zero. If any stochastic variables lack a random() method, PyMC will raise an exception. seed(): Same as draw_from_prior, but only stochastics whose rseed attribute is not None are changed. As introduced in the previous chapter, the helper function graph.dag produces graphical representations of models (see [Jordan2004]). Models have the following important attributes: • variables • nodes • stochastics • potentials • deterministics • observed_stochastics • containers • value • logp In addition, models expose each node they contain as an attribute. For instance, if model M were produced from model ((??)) M.s would return the switchpoint variable. 36 Chapter 5. Fitting Models PyMC Documentation, Release 2.3.5 5.3 Maximum a posteriori estimates The MAP class sets all stochastic variables to their maximum a posteriori values using functions in SciPy’s optimize package; hence, SciPy must be installed to use it. MAP can only handle variables whose dtype is float, so it will not work, for example, on model ((??)). To fit the model in examples/gelman_bioassay.py using MAP, do the following: >>> from pymc.examples import gelman_bioassay >>> M = pymc.MAP(gelman_bioassay) >>> M.fit() This call will cause 𝑀 to fit the model using modified Powell optimization, which does not require derivatives. The variables in DisasterModel have now been set to their maximum a posteriori values: >>> M.alpha.value array(0.8465892309923545) >>> M.beta.value array(7.7488499785334168) In addition, the AIC and BIC of the model are now available: >>> M.AIC 7.9648372671389458 >>> M.BIC 6.7374259893787265 MAP has two useful methods: fit(method=’fmin_powell’, iterlim=1000, tol=.0001): The optimization method may be fmin, fmin_l_bfgs_b, fmin_ncg, fmin_cg, or fmin_powell. See the documentation of SciPy’s optimize package for the details of these methods. The tol and iterlim parameters are passed to the optimization function under the appropriate names. revert_to_max(): If the values of the constituent stochastic variables change after fitting, this function will reset them to their maximum a posteriori values. If you’re going to use an optimization method that requires derivatives, MAP‘s __init__ method can take additional parameters eps and diff_order. diff_order, which must be an integer, specifies the order of the numerical approximation (see the SciPy function derivative). The step size for numerical derivatives is controlled by eps, which may be either a single value or a dictionary of values whose keys are variables (actual objects, not names). The useful attributes of MAP are: logp: The joint log-probability of the model. logp_at_max: The maximum joint log-probability of the model. AIC: Akaike’s information criterion for this model ([Akaike1973],[Burnham2002]_). BIC: The Bayesian information criterion for this model [Schwarz1978]. One use of the MAP class is finding reasonable initial states for MCMC chains. Note that multiple Model subclasses can handle the same collection of nodes. 5.4 Normal approximations The NormApprox class extends the MAP class by approximating the posterior covariance of the model using the Fisher information matrix, or the Hessian of the joint log probability at the maximum. To fit the model in examples/gelman_bioassay.py using NormApprox, do: 5.3. Maximum a posteriori estimates 37 PyMC Documentation, Release 2.3.5 >>> N = pymc.NormApprox(gelman_bioassay) >>> N.fit() The approximate joint posterior mean and covariance of the variables are available via the attributes mu and C: >>> N.mu[N.alpha] array([ 0.84658923]) >>> N.mu[N.alpha, N.beta] array([ 0.84658923, 7.74884998]) >>> N.C[N.alpha] matrix([[ 1.03854093]]) >>> N.C[N.alpha, N.beta] matrix([[ 1.03854093, 3.54601911], [ 3.54601911, 23.74406919]]) As with MAP, the variables have been set to their maximum a posteriori values (which are also in the mu attribute) and the AIC and BIC of the model are available. In addition, it’s now possible to generate samples from the posterior as with MCMC: >>> N.sample(100) >>> N.trace('alpha')[::10] array([-0.85001278, 1.58982854, 1.0388088 , 0.07626688, 1.15359581, -0.25211939, 1.39264616, 0.22551586, 2.69729987, 1.21722872]) >>> N.trace('beta')[::10] array([ 2.50203663, 14.73815047, 11.32166303, 0.43115426, 10.1182532 , 7.4063525 , 11.58584317, 8.99331152, 11.04720439, 9.5084239 ]) Any of the database backends can be used (chapter Saving and managing sampling results). In addition to the methods and attributes of MAP, NormApprox provides the following methods: sample(iter): Samples from the approximate posterior distribution are drawn and stored. isample(iter): An ‘interactive’ version of sample(): sampling can be paused, returning control to the user. draw: Sets all variables to random values drawn from the approximate posterior. It provides the following additional attributes: mu: A special dictionary-like object that can be keyed with multiple variables. N.mu[p1, p2, p3] would return the approximate posterior mean values of stochastic variables p1, p2 and p3, raveled and concatenated to form a vector. C: Another special dictionary-like object. N.C[p1, p2, p3] would return the approximate posterior covariance matrix of stochastic variables p1, p2 and p3. As with mu, these variables’ values are raveled and concatenated before their covariance matrix is constructed. 5.5 Markov chain Monte Carlo: the MCMC class The MCMC class implements PyMC’s core business: producing ‘traces’ for a model’s variables which, with careful thinning, can be considered independent joint samples from the posterior. See Tutorial for an example of basic usage. MCMC‘s primary job is to create and coordinate a collection of ‘step methods’, each of which is responsible for updating one or more variables. The available step methods are described below. Instructions on how to create your own step method are available in Extending PyMC. MCMC provides the following useful methods: 38 Chapter 5. Fitting Models PyMC Documentation, Release 2.3.5 sample(iter, burn, thin, tune_interval, tune_throughout, save_interval, ...): Runs the MCMC algorithm and produces the traces. The iter argument controls the total number of MCMC iterations. No tallying will be done during the first burn iterations; these samples will be forgotten. After this burn-in period, tallying will be done each thin iterations. Tuning will be done each tune_interval iterations. If tune_throughout=False, no more tuning will be done after the burnin period. The model state will be saved every save_interval iterations, if given. isample(iter, burn, thin, tune_interval, tune_throughout, save_interval, ...): An interactive version of sample. The sampling loop may be paused at any time, returning control to the user. use_step_method(method, *args, **kwargs): Creates an instance of step method class method to handle some stochastic variables. The extra arguments are passed to the init method of method. Assigning a step method to a variable manually will prevent the MCMC instance from automatically assigning one. However, you may handle a variable with multiple step methods. goodness(): Calculates goodness-of-fit (GOF) statistics according to [Brooks2000]. save_state(): Saves the current state of the sampler, including all stochastics, to the database. This allows the sampler to be reconstituted at a later time to resume sampling. This is not supported yet for the sqlite backend. restore_state(): Restores the sampler to the state stored in the database. stats(): Generate summary statistics for all nodes in the model. remember(trace_index): Set all variables’ values from frame trace_index in the database. MCMC samplers’ step methods can be accessed via the step_method_dict attribute. M.step_method_dict[x] returns a list of the step methods M will use to handle the stochastic variable x. After sampling, the information tallied by M can be queried via M.db.trace_names. In addition to the values of variables, tuning information for adaptive step methods is generally tallied. These ‘traces’ can be plotted to verify that tuning has in fact terminated. You can produce ‘traces’ for arbitrary functions with zero arguments as well. If you issue the command M._funs_to_tally[’trace_name’] = f before sampling begins, then each time the model variables’ values are tallied, f will be called with no arguments, and the return value will be tallied. After sampling ends you can retrieve the trace as M.trace[’trace_name’]. 5.6 The Sampler class MCMC is a subclass of a more general class called Sampler. Samplers fit models with Monte Carlo fitting methods, which characterize the posterior distribution by approximate samples from it. They are initialized as follows: Sampler(input=None, db=’ram’, name=’Sampler’, reinit_model=True, calc_deviance=False, verbose=0). The input argument is a module, list, tuple, dictionary, set, or object that contains all elements of the model, the db argument indicates which database backend should be used to store the samples (see chapter Saving and managing sampling results), reinit_model is a boolean flag that indicates whether the model should be re-initialised before running, and calc_deviance is a boolean flag indicating whether deviance should be calculated for the model at each iteration. Samplers have the following important methods: sample(iter, length, verbose, ...): Samples from the joint distribution. The iter argument controls how many times the sampling loop will be run, and the length argument controls the initial size of the database that will be used to store the samples. isample(iter, length, verbose, ...): The same as sample, but the sampling is done interactively: you can pause sampling at any point and be returned to the Python prompt to inspect progress and adjust fitting parameters. While sampling is paused, the following methods are useful: 5.6. The Sampler class 39 PyMC Documentation, Release 2.3.5 icontinue(): Continue interactive sampling. halt(): Truncate the database and clean up. tally(): Write all variables’ current values to the database. The actual write operation depends on the specified database backend. save_state(): Saves the current state of the sampler, including all stochastics, to the database. This allows the sampler to be reconstituted at a later time to resume sampling. This is not supported yet for the sqlite backend. restore_state(): Restores the sampler to the state stored in the database. stats(): Generate summary statistics for all nodes in the model. remember(trace_index): Set all variables’ values from frame trace_index in the database. Note that the trace_index is different from the current iteration, since not all samples are necessarily saved due to burning and thinning. In addition, the sampler attribute deviance is a deterministic variable valued as the model’s deviance at its current state. 5.7 Step methods Step method objects handle individual stochastic variables, or sometimes groups of them. They are responsible for making the variables they handle take single MCMC steps conditional on the rest of the model. Each subclass of StepMethod implements a method called step(), which is called by MCMC. Step methods with adaptive tuning parameters can optionally implement a method called tune(), which causes them to assess performance (based on the acceptance rates of proposed values for the variable) so far and adjust. The major subclasses of StepMethod are Metropolis, AdaptiveMetropolis and Slicer. PyMC provides several flavors of the basic Metropolis steps. There are Gibbs sampling (Gibbs) steps, but they are not ready for use as of the current release, but since it is feasible to write Gibbs step methods for particular applications, the Gibbs base class will be documented here. 5.7.1 Metropolis step methods Metropolis and subclasses implement Metropolis-Hastings steps. To tell an MCMC object 𝑀 to handle a variable x with a Metropolis step method, you might do the following: M.use_step_method(pymc.Metropolis, x, proposal_sd=1., proposal_distribution='Normal') Metropolis itself handles float-valued variables, and subclasses DiscreteMetropolis and BinaryMetropolis handle integer- and boolean-valued variables, respectively. Subclasses of Metropolis must implement the following methods: propose(): Sets the values of the variables handled by the Metropolis step method to proposed values. reject(): If the Metropolis-Hastings acceptance test fails, this method is called to reset the values of the variables to their values before propose() was called. Note that there is no accept() method; if a proposal is accepted, the variables’ values are simply left alone. Subclasses that use proposal distributions other than symmetric random-walk may specify the ‘Hastings factor’ by changing the hastings_factor method. See Extending PyMC for an example. Metropolis‘ __init__ method takes the following arguments: stochastic: The variable to handle. 40 Chapter 5. Fitting Models PyMC Documentation, Release 2.3.5 proposal_sd: A float or array of floats. This sets the proposal standard deviation if the proposal distribution is normal. scale: A float, defaulting to 1. If the scale argument is provided but not proposal_sd, proposal_sd is computed as follows: if all(self.stochastic.value != 0.): self.proposal_sd = ones(shape(self.stochastic.value)) * \ abs(self.stochastic.value) * scale else: self.proposal_sd = ones(shape(self.stochastic.value)) * scale proposal_distribution: A string indicating which distribution should be used for proposals. Current options are ’Normal’ and ’Prior’. If proposal_distribution=None, the proposal distribution is chosen automatically. It is set to ’Prior’ if the variable has no children and has a random method, and to ’Normal’ otherwise. verbose: An integer. By convention 0 indicates no output, 1 shows a progress bar only, 2 provides basic feedback about the current MCMC run, while 3 and 4 provide low and high debugging verbosity, respectively. Alhough the proposal_sd attribute is fixed at creation, Metropolis step methods adjust their initial proposal standard deviations using an attribute called adaptive_scale_factor. When tune() is called, the acceptance ratio of the step method is examined, and this scale factor is updated accordingly. If the proposal distribution is normal, proposals will have standard deviation self.proposal_sd * self.adaptive_scale_factor. By default, tuning will continue throughout the sampling loop, even after the burnin period is over. This can be changed via the tune_throughout argument to MCMC.sample. If an adaptive step method’s tally flag is set (the default for Metropolis), a trace of its tuning parameters will be kept. If you allow tuning to continue throughout the sampling loop, it is important to verify that the ‘Diminishing Tuning’ condition of [Roberts2007] is satisfied: the amount of tuning should decrease to zero, or tuning should become very infrequent. If a Metropolis step method handles an array-valued variable, it proposes all elements independently but simultaneously. That is, it decides whether to accept or reject all elements together but it does not attempt to take the posterior correlation between elements into account. The AdaptiveMetropolis class (see below), on the other hand, does make correlated proposals. 5.7.2 The AdaptiveMetropolis class The AdaptativeMetropolis (AM) step method works like a regular Metropolis step method, with the exception that its variables are block-updated using a multivariate jump distribution whose covariance is tuned during sampling. Although the chain is non-Markovian, it has correct ergodic properties (see [Haario2001]). To tell an MCMC object 𝑀 to handle variables x, y and 𝑧 with an AdaptiveMetropolis instance, you might do the following: M.use_step_method(pymc.AdaptiveMetropolis, [x,y,z], \ scales={'x':1, 'y':2, 'z':.5}, delay=10000) AdaptativeMetropolis‘s init method takes the following arguments: stochastics: The stochastic variables to handle. These will be updated jointly. cov (optional): An initial covariance matrix. Defaults to the identity matrix, adjusted according to the scales argument. delay (optional): The number of iterations to delay before computing the empirical covariance matrix. scales (optional): The initial covariance matrix will be diagonal, and its diagonal elements will be set to scales times the stochastics’ values, squared. 5.7. Step methods 41 PyMC Documentation, Release 2.3.5 interval (optional): The number of iterations between updates of the covariance matrix. Defaults to 1000. greedy (optional): If True, only accepted jumps will be counted toward the delay before the covariance is first computed. Defaults to True. verbose: An integer from 0 to 4 controlling the verbosity of the step method’s printed output. shrink_if_necessary (optional): Whether the proposal covariance should be shrunk if the acceptance rate becomes extremely small. In this algorithm, jumps are proposed from a multivariate normal distribution with covariance matrix Σ. The algorithm first iterates until delay samples have been drawn (if greedy is true, until delay jumps have been accepted). At this point, Σ is given the value of the empirical covariance of the trace so far and sampling resumes. The covariance is then updated each interval iterations throughout the entire sampling run 1 . It is this constant adaptation of the proposal distribution that makes the chain non-Markovian. 5.7.3 The DiscreteMetropolis class This class is just like Metropolis, but specialized to handle Stochastic instances with dtype int. The jump proposal distribution can either be ’Normal’, ’Prior’ or ’Poisson’ (the default). In the normal case, the proposed value is drawn from a normal distribution centered at the current value and then rounded to the nearest integer. 5.7.4 The BinaryMetropolis class This class is specialized to handle Stochastic instances with dtype bool. For array-valued variables, BinaryMetropolis can be set to propose from the prior by passing in dist="Prior". Otherwise, the argument p_jump of the init method specifies how probable a change is. Like Metropolis‘ attribute proposal_sd, p_jump is tuned throughout the sampling loop via adaptive_scale_factor. For scalar-valued variables, BinaryMetropolis behaves like a Gibbs sampler, since this requires no additional expense. The p_jump and adaptive_scale_factor parameters are not used in this case. 5.7.5 The Slicer class The Slicer class implements Slice sampling ([Neal2003]). To tell an MCMC object 𝑀 to handle a variable x with a Slicer step method, you might do the following: M.use_step_method(pymc.Slicer, x, w=10, m=10000, doubling=True) Slicer‘s init method takes the following arguments: stochastics: The stochastic variables to handle. These will be updated jointly. w (optional): The initial width of the horizontal slice (Defaults to 1). This will be updated via either stepping-out or doubling procedures. m (optional): The multiplier defining the maximum slice size as 𝑚𝑤 (Defaults to 1000). tune (optional): A flag indicating whether to tune the initial slice width (Defaults to True). doubling (optional): A flag for using doubling procedure instead of stepping out (Defaults to False) tally (optional): Flag for recording values for trace (Defaults to True). 1 The covariance is estimated recursively from the previous value and the last interval samples, instead of computing it each time from the entire trace. 42 Chapter 5. Fitting Models PyMC Documentation, Release 2.3.5 verbose: An integer from -1 to 4 controlling the verbosity of the step method’s printed output (Defaults to -1). The *slice sampler* generates posterior samples by alternately drawing “slices” from the vertical (y) and horizontal (x) planes of a distribution. It first samples from the conditional distribution for y given some current value of x, which is uniform over the (0, 𝑓 (𝑥)). Conditional on this value for y, it then samples x, which is uniform on 𝑆 = 𝑥 : 𝑦 < 𝑓 (𝑥); that is the “slice” defined by the y value. Hence, this algorithm automatically adapts its to the local characteristics of the posterior. The steps required to perform a single iteration of the slice sampler to update the current value of 𝑥𝑖 is as follows: 1. Sample y uniformly on (0, 𝑓 (𝑥𝑖 )). 2. Use this value y to define a horizontal slice 𝑆 = {𝑥 : 𝑦 < 𝑓 (𝑥)}. 3. Establish an interval, 𝐼 = (𝑥𝑎 , 𝑥𝑏 ), around 𝑥𝑖 that contains most of the slice. 4. Sample 𝑥𝑖+1 from the region of the slice overlaping I. Hence, slice sampling employs an auxilliary variable (y) that is not retained at the end of the iteration. Note that in practice one may operate on the log scale such that 𝑔(𝑥) = log(𝑓 (𝑥)) to avoid floating-point underflow. In this case, the auxiliary variable becomes 𝑧 = 𝑙𝑜𝑔(𝑦) = 𝑔(𝑥𝑖 )𝑒, where 𝑒 ∼ Exp(1), resulting in the slice 𝑆 = {𝑥 : 𝑧 < 𝑔(𝑥)}. There are many ways of establishing and sampling from the interval I, with the only restriction being that the resulting Markov chain leaves 𝑓 (𝑥) invariant. The objective is to include as much of the slice as possible, so that the potential step size can be large, but not (much) larger than the slice, so that the sampling of invalid points is minimized. Ideally, we would like it to be the slice itself, but it may not always be feasible to determine (and certainly not automatically). One method for determining a sampling interval for 𝑥𝑖+1 involves specifying an initial “guess” at the slice width w, and iteratively moving the endpoints out (growing the interval) until either (1) the interval reaches a maximum prespecified width or (2) y is less than the 𝑓 (𝑥) evaluated both at the left and the right interval endpoints. This is the stepping out method. The efficiency of stepping out depends largely on the ability to pick a reasonable interval w from which to sample. Otherwise, the doubling procedure may be preferable, as it can be expanded faster. It simply doubles the size of the interval until both endpoints are outside the slice. 5.8 Gibbs step methods Gibbs step methods handle conjugate submodels. These models usually have two components: the ‘parent’ and the ‘children’. For example, a gamma-distributed variable serving as the precision of several normally-distributed variables is a conjugate submodel; the gamma variable is the parent and the normal variables are the children. This section describes PyMC’s current scheme for Gibbs step methods, several of which are in a semi-working state in the sandbox. It is meant to be as generic as possible to minimize code duplication, but it is admittedly complicated. Feel free to subclass StepMethod directly when writing Gibbs step methods if you prefer. Gibbs step methods that subclass PyMC’s Gibbs should define the following class attributes: child_class: The class of the children in the submodels the step method can handle. parent_class: The class of the parent. parent_label: The label the children would apply to the parent in a conjugate submodel. In the gamma-normal example, this would be tau. linear_OK: A flag indicating whether the children can use linear combinations involving the parent as their actual parent without destroying the conjugacy. A subclass of Gibbs that defines these attributes only needs to implement a propose() method, which will be called by Gibbs.step(). The resulting step method will be able to handle both conjugate and ‘non-conjugate’ cases. The conjugate case corresponds to an actual conjugate submodel. In the nonconjugate case all the children are 5.8. Gibbs step methods 43 PyMC Documentation, Release 2.3.5 of the required class, but the parent is not. In this case the parent’s value is proposed from the likelihood and accepted based on its prior. The acceptance rate in the nonconjugate case will be less than one. The inherited class method Gibbs.competence will determine the new step method’s ability to handle a variable x by checking whether: • all x‘s children are of class child_class, and either apply parent_label to x directly or (if linear_OK=True) to a LinearCombination object (Building models), one of whose parents contains x. • x is of class parent_class If both conditions are met, pymc.conjugate_Gibbs_competence will be returned. If only the first is met, pymc.nonconjugate_Gibbs_competence will be returned. 5.8.1 Granularity of step methods: one-at-a-time vs. block updating There is currently no way for a stochastic variable to compute individual terms of its log-probability; it is computed all together. This means that updating the elements of a array-valued variable individually would be inefficient, so all existing step methods update array-valued variables together, in a block update. To update an array-valued variable’s elements individually, simply break it up into an array of scalar-valued variables. Instead of this: A = pymc.Normal('A', value=zeros(100), mu=0., tau=1.) do this: A = [pymc.Normal('A_%i'%i, value=0., mu=0., tau=1.) for i in range(100)] An individual step method will be assigned to each element of A in the latter case, and the elements will be updated individually. Note that A can be broken up into larger blocks if desired. 5.8.2 Automatic assignment of step methods Every step method subclass (including user-defined ones) that does not require any __init__ arguments other than the stochastic variable to be handled adds itself to a list called StepMethodRegistry in the PyMC namespace. If a stochastic variable in an MCMC object has not been explicitly assigned a step method, each class in StepMethodRegistry is allowed to examine the variable. To do so, each step method implements a class method called competence(stochastic), whose only argument is a single stochastic variable. These methods return values from 0 to 3; 0 meaning the step method cannot safely handle the variable and 3 meaning it will most likely perform well for variables like this. The MCMC object assigns the step method that returns the highest competence value to each of its stochastic variables. 44 Chapter 5. Fitting Models CHAPTER 6 Saving and managing sampling results 6.1 Accessing Sampled Data The recommended way to access data from an MCMC run, irrespective of the database backend, is to use the trace method: >>> from pymc.examples import disaster_model >>> from pymc import MCMC >>> M = MCMC(disaster_model) >>> M.sample(10) Sampling: 100% [00000000000000000000000000000000000000000000000000] Iterations: 10 >>> M.trace('early_mean')[:] array([ 2.28320992, 2.28320992, 2.28320992, 2.28320992, 2.28320992, 2.36982455, 2.36982455, 3.1669422 , 3.1669422 , 3.14499489]) M.trace(’early_mean’) returns a copy of the Trace instance associated with the tallyable object early_mean: >>> M.trace('early_mean') <pymc.database.ram.Trace object at 0x7fa4877a8b50> Particular subsamples from the trace are obtained using the slice notation [], similar to NumPy arrays. By default, trace returns the samples from the last chain. To return the samples from all the chains, use chain=None: >>> M.sample(5) Sampling: 100% [000000000000000000000000000000000000000000000000000] Iterations: 5 >>> M.trace('early_mean', chain=None)[:] array([ 2.28320992, 2.28320992, 2.28320992, 2.28320992, 2.28320992, 2.36982455, 2.36982455, 3.1669422 , 3.1669422 , 3.14499489, 3.14499489, 3.14499489, 3.14499489, 2.94672454, 3.10767686]) 6.1.1 Output Summaries PyMC samplers include a couple of methods that are useful for obtaining summaries of the model, or particular member nodes, rather than the entire trace. The summary method can be used to generate a pretty-printed summary of posterior quantities. For example, if we want a statistical snapshot of the early_mean node: >>> M.early_mean.summary() early_mean: Mean SD MC Error 95% HPD interval ------------------------------------------------------------------ 45 PyMC Documentation, Release 2.3.5 3.075 0.287 0.01 [ 2.594 3.722] Posterior quantiles: 2.5 25 50 75 97.5 |---------------|===============|===============|---------------| 2.531 2.876 3.069 3.255 3.671 A method of the same name exists for the sampler, which yields summaries for every node in the model. Alternatively, we may wish to write posterior statistics to a file, where they may be imported into a spreadsheet or plotting package. In that case, write_csv may be called to generate a comma-separated values (csv) file containing all available statistics for each node: M.write_csv("disasters.csv", variables=["early_mean", "late_mean", "switchpoint"]) Fig. 6.1: Summary statistics of stochastics from the disaster_model example, shown in a spreadsheet. write_csv is called with a single mandatory argument, the name of the output file for the summary statistics, and several optional arguments, including a list of parameters for which summaries are desired (if not given, all model nodes are summarized) and an alpha level for calculating credible intervalse (defaults to 0.05). 6.2 Saving Data to Disk By default, the database backend selected by the MCMC sampler is the ram backend, which simply holds the data in memory. Now, we will create a sampler that instead will write data to a pickle file: >>> M = MCMC(disaster_model, db='pickle', dbname='Disaster.pickle') >>> M.db <pymc.database.pickle.Database object at 0x7fa486623d90> >>> M.sample(10) >>> M.db.close() Note that in this particular case, no data is written to disk before the call to db.close. The close method will flush data to disk and prepare the database for a potential session exit. Closing a Python session without calling close beforehand is likely to corrupt the database, making the data irretrievable. To simply flush data to disk without closing the database, use the commit method. Some backends not only have the ability to store the traces, but also the state of the step methods at the end of sampling. This is particularly useful when long warm-up periods are needed to tune the jump parameters. When the database is loaded in a new session, the step methods query the database to fetch the state they were in at the end of the last trace. Check that you close() the database before closing the Python session. 6.3 Reloading a Database To load a file created in a previous session, use the load function from the backend: 46 Chapter 6. Saving and managing sampling results PyMC Documentation, Release 2.3.5 >>> db = pymc.database.pickle.load('Disaster.pickle') >>> len(db.trace('early_mean')[:]) 10 The db object also has a trace method identical to that of Sampler. You can hence inspect the results of a model, even when you don’t have the model around. To add a new trace to this file, we need to create an MCMC instance. This time, instead of setting db=’pickle’, we will pass the existing Database instance and sample as usual. A new trace will be appended to the first: >>> M = MCMC(disaster_model, db=db) >>> M.sample(5) Sampling: 100% [000000000000000000000000000000000000000000000000000] Iterations: 5 >>> len(M.trace('early_model', chain=None)[:]) 15 >>> M.db.close() 6.3.1 The ram backend Used by default, this backend simply holds a copy in memory, with no output written to disk. This is useful for short runs or testing. For long runs generating large amount of data, using this backend may fill the available memory, forcing the OS to store data in the cache, slowing down all other applications. 6.3.2 The no_trace backend This backend simply does not store the trace. This may be useful for testing purposes. 6.3.3 The txt backend With the txt backend, the data is written to disk in ASCII files. More precisely, the dbname argument is used to create a top directory into which chain directories, called Chain_<#>, are going to be created each time sample is called: dbname/ Chain_0/ <object0 <object1 ... Chain_1/ <object0 <object1 ... ... name>.txt name>.txt name>.txt name>.txt In each one of these chain directories, files named <variable name>.txt are created, storing the values of the variable as rows of text: # Variable: e # Sample shape: (5,) # Date: 2008-11-18 17:19:13.554188 3.033672373807017486e+00 3.033672373807017486e+00 ... 6.3. Reloading a Database 47 PyMC Documentation, Release 2.3.5 While the txt backend makes it easy to load data using other applications and programming languages, it is not optimized for speed nor memory efficiency. If you plan on generating and handling large datasets, read on for better options. 6.3.4 The pickle backend The pickle database relies on the cPickle module to save the traces. Use of this backend is appropriate for smallscale, short-lived projects. For longer term or larger projects, the pickle backend should be avoided since generated files might be unreadable across different Python versions. The pickled file is a simple dump of a dictionary containing the NumPy arrays storing the traces, as well as the state of the Sampler‘s step methods. 6.3.5 The sqlite backend The sqlite backend is based on the python module sqlite3 (built-in to Python versions greater than 2.4) . It opens an SQL database named dbname, and creates one table per tallyable objects. The rows of this table store a key, the chain index and the values of the objects: key (INTT), trace (INT), v1 (FLOAT), v2 (FLOAT), v3 (FLOAT) ... The key is autoincremented each time a new row is added to the table, that is, each time tally is called by the sampler. Note that the savestate feature is not implemented, that is, the state of the step methods is not stored internally in the database. 6.3.6 The hdf5 backend The hdf5 backend uses pyTables to save data in binary HDF5 format. The hdf5 database is fast and can store huge traces, far larger than the available RAM. Data can be compressed and decompressed on the fly to reduce the disk footprint. Another feature of this backends is that it can store arbitrary objects. Whereas the other backends are limited to numerical values, hdf5 can tally any object that can be pickled, opening the door for powerful and exotic applications (see pymc.gp). The internal structure of an HDF5 file storing both numerical values and arbitrary objects is as follows: / (root) /chain0/ (Group) 'Chain #0' /chain0/PyMCSamples (Table(N,)) 'PyMC Samples' /chain0/group0 (Group) 'Group storing objects.' /chain0/group0/<object0 name> (VLArray(N,)) '<object0 name> samples.' /chain0/group0/<object1 name> (VLArray(N,)) '<object1 name> samples.' ... /chain1/ (Group) 'Chain #1' ... All standard numerical values are stored in a Table, while objects are stored in individual VLArrays. The hdf5 Database takes the following parameters: • dbname (string) Name of the hdf5 file. • dbmode (string) File mode: a: append, w: overwrite, r: read-only. • dbcomplevel (int (0-9)) Compression level, 0: no compression. • dbcomplib (string) Compression library (zlib, bzip2, lzo) 48 Chapter 6. Saving and managing sampling results PyMC Documentation, Release 2.3.5 According the the pyTables manual, zlib ([Roelofs2010]) has a fast decompression, relatively slow compression, and a good compression ratio. LZO ([Oberhumer2008]) has a fast compression, but a low compression ratio. bzip2 ([Seward2007]) has an excellent compression ratio but requires more CPU. Note that some of these compression algorithms require additional software to work (see the pyTables manual). 6.4 Writing a New Backend It is relatively easy to write a new backend for PyMC. The first step is to look at the database.base module, which defines barebone Database and Trace classes. This module contains documentation on the methods that should be defined to get a working backend. Testing your new backend should be trivial, since the test_database module contains a generic test class that can easily be subclassed to check that the basic features required of all backends are implemented and working properly. 6.4. Writing a New Backend 49 PyMC Documentation, Release 2.3.5 50 Chapter 6. Saving and managing sampling results CHAPTER 7 Model checking and diagnostics 7.1 Convergence Diagnostics Valid inferences from sequences of MCMC samples are based on the assumption that the samples are derived from the true posterior distribution of interest. Theory guarantees this condition as the number of iterations approaches infinity. It is important, therefore, to determine the minimum number of samples required to ensure a reasonable approximation to the target posterior density. Unfortunately, no universal threshold exists across all problems, so convergence must be assessed independently each time MCMC estimation is performed. The procedures for verifying convergence are collectively known as convergence diagnostics. One approach to analyzing convergence is analytical, whereby the variance of the sample at different sections of the chain are compared to that of the limiting distribution. These methods use distance metrics to analyze convergence, or place theoretical bounds on the sample variance, and though they are promising, they are generally difficult to use and are not prominent in the MCMC literature. More common is a statistical approach to assessing convergence. With this approach, rather than considering the properties of the theoretical target distribution, only the statistical properties of the observed chain are analyzed. Reliance on the sample alone restricts such convergence criteria to heuristics. As a result, convergence cannot be guaranteed. Although evidence for lack of convergence using statistical convergence diagnostics will correctly imply lack of convergence in the chain, the absence of such evidence will not guarantee convergence in the chain. Nevertheless, negative results for one or more criteria may provide some measure of assurance to users that their sample will provide valid inferences. For most simple models, convergence will occur quickly, sometimes within a the first several hundred iterations, after which all remaining samples of the chain may be used to calculate posterior quantities. For more complex models, convergence requires a significantly longer burn-in period; sometimes orders of magnitude more samples are needed. Frequently, lack of convergence will be caused by poor mixing (Figure 7.1). Recall that mixing refers to the degree to which the Markov chain explores the support of the posterior distribution. Poor mixing may stem from inappropriate proposals (if one is using the Metropolis-Hastings sampler) or from attempting to estimate models with highly correlated variables. 7.1.1 Informal Methods The most straightforward approach for assessing convergence is based on simply plotting and inspecting traces and histograms of the observed MCMC sample. If the trace of values for each of the stochastics exhibits asymptotic behavior 1 over the last 𝑚 iterations, this may be satisfactory evidence for convergence. A similar approach involves plotting a histogram for every set of 𝑘 iterations (perhaps 50-100) beyond some burn in threshold 𝑛; if the histograms are not visibly different among the sample intervals, this is reasonable evidence for convergence. Note that such diagnostics should be carried out for each stochastic estimated by the MCMC algorithm, because convergent behavior by one variable does not imply evidence for convergence for other variables in the analysis. An extension of this 1 Asymptotic behaviour implies that the variance and the mean value of the sample stays relatively constant over some arbitrary period. 51 PyMC Documentation, Release 2.3.5 Fig. 7.1: An example of a poorly-mixing sample in two dimensions. Notice that the chain is trapped in a region of low probability relative to the mean (dot) and variance (oval) of the true posterior quantity. 52 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 approach can be taken when multiple parallel chains are run, rather than just a single, long chain. In this case, the final values of 𝑐 chains run for 𝑛 iterations are plotted in a histogram; just as above, this is repeated every 𝑘 iterations thereafter, and the histograms of the endpoints are plotted again and compared to the previous histogram. This is repeated until consecutive histograms are indistinguishable. Another ad hoc method for detecting lack of convergence is to examine the traces of several MCMC chains initialized with different starting values. Overlaying these traces on the same set of axes should (if convergence has occurred) show each chain tending toward the same equilibrium value, with approximately the same variance. Recall that the tendency for some Markov chains to converge to the true (unknown) value from diverse initial values is called ergodicity. This property is guaranteed by the reversible chains constructed using MCMC, and should be observable using this technique. Again, however, this approach is only a heuristic method, and cannot always detect lack of convergence, even though chains may appear ergodic. Fig. 7.2: An example of metastability in a two-dimensional parameter space. The chain appears to be stable in one region of the parameter space for an extended period, then unpredictably jumps to another region of the space. A principal reason that evidence from informal techniques cannot guarantee convergence is a phenomenon called metastability. Chains may appear to have converged to the true equilibrium value, displaying excellent qualities by any of the methods described above. However, after some period of stability around this value, the chain may suddenly move to another region of the parameter space (Figure 7.2). This period of metastability can sometimes be very long, and therefore escape detection by these convergence diagnostics. Unfortunately, there is no statistical technique available for detecting metastability. 7.1.2 Formal Methods Along with the ad hoc techniques described above, a number of more formal methods exist which are prevalent in the literature. These are considered more formal because they are based on existing statistical methods, such as time series analysis. 7.1. Convergence Diagnostics 53 PyMC Documentation, Release 2.3.5 PyMC currently includes three formal convergence diagnostic methods. The first, proposed by [Geweke1992], is a time-series approach that compares the mean and variance of segments from the beginning and end of a single chain. 𝑧 = √︀ 𝜃¯𝑎 − 𝜃¯𝑏 𝑉 𝑎𝑟(𝜃𝑎 ) + 𝑉 𝑎𝑟(𝜃𝑏 ) where 𝑎 is the early interval and 𝑏 the late interval. If the z-scores (theoretically distributed as standard normal variates) of these two segments are similar, it can provide evidence for convergence. PyMC calculates z-scores of the difference between various initial segments along the chain, and the last 50% of the remaining chain. If the chain has converged, the majority of points should fall within 2 standard deviations of zero. Diagnostic z-scores can be obtained by calling the geweke function. It accepts either (1) a single trace, (2) a Node or Stochastic object, or (4) an entire Model object: geweke(x, first=0.1, last=0.5, intervals=20) The arguments expected are the following: • pymc_object: The object that is or contains the output trace(s). • first (optional): First portion of chain to be used in Geweke diagnostic. Defaults to 0.1 (i.e. first 10% of chain). • last (optional): Last portion of chain to be used in Geweke diagnostic. Defaults to 0.5 (i.e. last 50% of chain). • intervals (optional): Number of sub-chains to analyze. Defaults to 20. The resulting scores are best interpreted graphically, using the geweke_plot function. This displays the scores in series, in relation to the 2 standard deviation boundaries around zero. Hence, it is easy to see departures from the standard normal assumption. geweke_plot takes either a single set of scores, or a dictionary of scores (output by geweke when an entire Sampler is passed) as its argument: def geweke_plot(scores, name='geweke', format='png', suffix='-diagnostic', path='./', fontmap = {1:10, 2:8, 3:6, 4:5, 5:4}, verbose=1) The arguments are defined as: • scores: The object that contains the Geweke scores. Can be a list (one set) or a dictionary (multiple sets). • name (optional): Name used for output files. For multiple scores, the dictionary keys are used as names. • format (optional): Graphic output file format (defaults to png). • suffix (optional): Suffix to filename (defaults to -diagnostic) • path (optional): The path for output graphics (defaults to working directory). • fontmap (optional): Dictionary containing the font map for the labels of the graphic. • verbose (optional): Verbosity level for output (defaults to 1). To illustrate, consider the sample model gelman_bioassay that is used to instantiate a MCMC sampler. The sampler is then run for a given number of iterations: >>> from pymc.examples import gelman_bioassay >>> S = pymc.MCMC(gelman_bioassay) >>> S.sample(10000, burn=5000) It is easiest simply to pass the entire sampler S the geweke function: >>> scores = pymc.geweke(S, intervals=20) >>> pymc.Matplot.geweke_plot(scores) 54 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 Fig. 7.3: Sample plot of Geweke z-scores for a variable using geweke_plot. The occurrence of the scores well within 2 standard deviations of zero gives not indicate of lack of convergence (top), while deviations exceeding 2 standard deviations suggest that additional samples are requred to achieve convergence (bottom). 7.1. Convergence Diagnostics 55 PyMC Documentation, Release 2.3.5 Alternatively, individual stochastics within S can be analyzed for convergence: >>> trace = S.trace("alpha")[:] >>> alpha_scores = pymc.geweke(trace, intervals=20) >>> pymc.Matplot.geweke_plot(alpha_scores, "alpha") An example of convergence and non-convergence of a chain using geweke_plot is given in Figure 7.3. The second diagnostic provided by PyMC is the [Raftery1995a] procedure. This approach estimates the number of iterations required to reach convergence, along with the number of burn-in samples to be discarded and the appropriate thinning interval. A separate estimate of both quantities can be obtained for each variable in a given model. As the criterion for determining convergence, the Raftery and Lewis approach uses the accuracy of estimation of a user-specified quantile. For example, we may want to estimate the quantile 𝑞 = 0.975 to within 𝑟 = 0.005 with probability 𝑠 = 0.95. In other words, 𝑃 𝑟(|ˆ 𝑞 − 𝑞| ≤ 𝑟) = 𝑠 From any sample of 𝜃, one can construct a binary chain: 𝑍 (𝑗) = 𝐼(𝜃(𝑗) ≤ 𝑢𝑞 ) where 𝑢𝑞 is the quantile value and 𝐼 is the indicator function. While {𝜃(𝑗) } is a Markov chain, {𝑍 (𝑗) } is not necessarily so. In any case, the serial dependency among 𝑍 (𝑗) decreases as the thinning interval 𝑘 increases. A value of 𝑘 is chosen to be the smallest value such that the first order Markov chain is preferable to the second order Markov chain. This thinned sample is used to determine number of burn-in samples. This is done by comparing the remaining samples from burn-in intervals of increasing length to the limiting distribution of the chain. An appropriate value is one for which the truncated sample’s distribution is within 𝜖 (arbitrarily small) of the limiting distribution. See [Raftery1995a] or [Gamerman1997] for computational details. Estimates for sample size tend to be conservative. This diagnostic is best used on a short pilot run of a particular model, and the results used to parameterize a subsequent sample that is to be used for inference. Its calling convention is as follows: raftery_lewis(x, q, r, s=.95, epsilon=.001, verbose=1) The arguments are: • pymc_object: The object that contains the Geweke scores. Can be a list (one set) or a dictionary (multiple sets). • q: Desired quantile to be estimated. • r: Desired accuracy for quantile. • s (optional): Probability of attaining the requested accuracy (defaults to 0.95). • epsilon (optional) : Half width of the tolerance interval required for the q-quantile (defaults to 0.001). • verbose (optional) : Verbosity level for output (defaults to 1). The code for raftery_lewis is based on the FORTRAN program gibbsit ([Raftery1995b]). For example, consider again a sampler S run for some model my_model: >>> S = pymc.MCMC(my_model) >>> S.sample(10000, burn=5000) One can pass either the entire sampler S or any stochastic within S to the raftery_lewis function, along with suitable arguments. Here, we have chosen 𝑞 = 0.025 (the lower limit of the equal-tailed 95% interval) and error 𝑟 = 0.01: >>> pymc.raftery_lewis(S, q=0.025, r=0.01) 56 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 This yields diagnostics as follows for each stochastic of S, as well as a dictionary containing the diagnostic quantities: ======================== Raftery-Lewis Diagnostic ======================== 937 iterations required (assuming independence) to achieve 0.01 accuracy with 95 percent probability. Thinning factor of 1 required to produce a first-order Markov chain. 39 iterations to be discarded at the beginning of the simulation (burn-in). 11380 subsequent iterations required. Thinning factor of 11 required to produce an independence chain. The third convergence diagnostic provided by PyMC is the Gelman-Rubin statistic ([Gelman1992]). This diagnostic uses multiple chains to check for lack of convergence, and is based on the notion that if multiple chains have converged, by definition they should appear very similar to one another; if not, one or more of the chains has failed to converge. The Gelman-Rubin diagnostic uses an analysis of variance approach to assessing convergence. That is, it calculates both the between-chain varaince (B) and within-chain varaince (W), and assesses whether they are different enough to worry about convergence. Assuming 𝑚 chains, each of length 𝑛, quantities are calculated by: 𝑚 𝑛 ∑︁ ¯ (𝜃.𝑗 − 𝜃¯.. )2 𝑚 − 1 𝑗=1 [︃ ]︃ 𝑚 𝑛 1 ∑︁ 1 ∑︁ 2 ¯ (𝜃𝑖𝑗 − 𝜃.𝑗 ) 𝑊 = 𝑚 𝑗=1 𝑛 − 1 𝑖=1 𝐵= for each scalar estimand 𝜃. Using these values, an estimate of the marginal posterior variance of 𝜃 can be calculated: 1 𝑛−1 ˆ 𝑊+ 𝐵 Var(𝜃|𝑦) = 𝑛 𝑛 Assuming 𝜃 was initialized to arbitrary starting points in each chain, this quantity will overestimate the true marginal posterior variance. At the same time, 𝑊 will tend to underestimate the within-chain variance early in the sampling run. However, in the limit as 𝑛 → ∞, both quantities will converge to the true variance of 𝜃. In light of this, the Gelman-Rubin statistic monitors convergence using the ratio: √︃ ˆ ˆ = Var(𝜃|𝑦) 𝑅 𝑊 This is called the potential scale reduction, since it is an estimate of the potential reduction in the scale of 𝜃 as the ˆ close to one (say, less than 1.1) to be number of simulations tends to infinity. In practice, we look for values of 𝑅 ˆ for each confident that a particular estimand has converged. In PyMC, the function gelman_rubin will calculate 𝑅 stochastic node in the passed model: >>> pymc.gelman_rubin(S) {'alpha': 1.0036389589627821, 'beta': 1.001503957313336, 'theta': [1.0013923468783055, 1.0274479503713816, 0.95365716267969636, 1.00267321019079]} For the best results, each chain should be initialized to highly dispersed starting values for each stochastic node. 7.1. Convergence Diagnostics 57 PyMC Documentation, Release 2.3.5 Fig. 7.4: Summary plot of parameters from gelman_bioassay model, showing credible intervals on the left and the Gelman-Rubin statistic on the right. 58 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 ˆ values will be plotted By default, when calling the summary_plot function using nodes with multiple chains, the 𝑅 alongside the posterior intervals. Additional convergence diagnostics are available in the R statistical package ([R2010]), via the CODA module ([Plummer2008]). PyMC includes a method coda for exporting model traces in a format that may be directly read by coda: >>> pymc.utils.coda(S) Generating CODA output ================================================== Processing deaths Processing beta Processing theta Processing alpha The lone argument is the PyMC sampler for which output is desired. Calling coda yields a file containing raw trace values (suffix .out) and a file containing indices to the trace values (suffix .ind). 7.2 Autocorrelation Plots Samples from MCMC algorithms are ususally autocorrelated, due partly to the inherent Markovian dependence structure. The degree of autocorrelation can be quantified using the autocorrelation function: Cov(𝑋𝑡 , 𝑋𝑡+𝑘 ) 𝜌𝑘 = √︀ Var(𝑋𝑡 )Var(𝑋𝑡+𝑘 ) 𝐸[(𝑋𝑡 − 𝜃)(𝑋𝑡+𝑘 − 𝜃)] = √︀ 𝐸[(𝑋𝑡 − 𝜃)2 ]𝐸[(𝑋𝑡+𝑘 − 𝜃)2 ] PyMC includes a function for plotting the autocorrelation function for each stochastics in the sampler (Figure 7.5). This allows users to examine the relationship among successive samples within sampled chains. Significant autocorrelation suggests that chains require thinning prior to use of the posterior statistics for inference. autocorrelation(pymc_object, name, maxlag=100, format='png', suffix='-acf', path='./', fontmap = {1:10, 2:8, 3:6, 4:5, 5:4}, verbose=1) • pymc_object: The object that is or contains the output trace(s). • name: Name used for output files. • maxlag: The highest lag interval for which autocorrelation is calculated. • format (optional): Graphic output file format (defaults to png). • suffix (optional): Suffix to filename (defaults to -diagnostic) • path (optional): The path for output graphics (defaults to working directory). • fontmap (optional): Dictionary containing the font map for the labels of the graphic. • verbose (optional): Verbosity level for output (defaults to 1). Autocorrelation plots can be obtained simply by passing the sampler to the autocorrelation function (within the Matplot module) directly: >>> S = pymc.MCMC(gelman_bioassay) >>> S.sample(10000, burn=5000) >>> pymc.Matplot.autocorrelation(S) 7.2. Autocorrelation Plots 59 PyMC Documentation, Release 2.3.5 Alternatively, variables within a model can be plotted individually. For example, the parameter beta that was estimated using sampler S for the gelman_bioassay model will yield a correlation plot as follows: >>> pymc.Matplot.autocorrelation(S.beta) Fig. 7.5: Sample autocorrelation plot for the switchpoint variable from the coal mining disasters example model. 7.3 Goodness of Fit Checking for model convergence is only the first step in the evaluation of MCMC model outputs. It is possible for an entirely unsuitable model to converge, so additional steps are needed to ensure that the estimated model adequately fits the data. One intuitive way of evaluating model fit is to compare model predictions with the observations used to fit the model. In other words, the fitted model can be used to simulate data, and the distribution of the simulated data should resemble the distribution of the actual data. Fortunately, simulating data from the model is a natural component of the Bayesian modelling framework. Recall, from the discussion on imputation of missing data, the posterior predictive distribution: ∫︁ 𝑝(˜ 𝑦 |𝑦) = 𝑝(˜ 𝑦 |𝜃)𝑓 (𝜃|𝑦)𝑑𝜃 Here, 𝑦˜ represents some hypothetical new data that would be expected, taking into account the posterior uncertainty in the model parameters. Sampling from the posterior predictive distribution is easy in PyMC. The code looks identical to the corresponding data stochastic, with two modifications: (1) the node should be specified as deterministic and (2) the statistical likelihoods should be replaced by random number generators. As an example, consider a simple dose-response model, where deaths are modeled as a binomial random variable for which the probability of death is a logit-linear function of the dose of a particular drug: n = [5]*4 dose = [-.86,-.3,-.05,.73] x = [0,1,3,5] alpha = pymc.Normal('alpha', mu=0.0, tau=0.01) 60 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 beta = pymc.Normal('beta', mu=0.0, tau=0.01) @pymc.deterministic def theta(a=alpha, b=beta, d=dose): """theta = inv_logit(a+b)""" return pymc.invlogit(a+b*d) # deaths ~ binomial(n, p) deaths = pymc.Binomial('deaths', n=n, p=theta, value=x, observed=True) The posterior predictive distribution of deaths uses the same functional form as the data likelihood, in this case a binomial stochastic. Here is the corresponding sample from the posterior predictive distribution: deaths_sim = pymc.Binomial('deaths_sim', n=n, p=theta) Notice that the observed stochastic pymc.Binomial has been replaced with a stochastic node that is identical in every respect to deaths, except that its values are not fixed to be the observed data – they are left to vary according to the values of the fitted parameters. The degree to which simulated data correspond to observations can be evaluated in at least two ways. First, these quantities can simply be compared visually. This allows for a qualitative comparison of model-based replicates and observations. If there is poor fit, the true value of the data may appear in the tails of the histogram of replicated data, while a good fit will tend to show the true data in high-probability regions of the posterior predictive distribution (Figure 7.6). Fig. 7.6: Data sampled from the posterior predictive distribution of a binomial random variate. The observed value (1) is shown by the dotted red line. The Matplot package in PyMC provides an easy way of producing such plots, via the gof_plot function. To illustrate, consider a single data point x and an array of values x_sim sampled from the posterior predictive distribution. The histogram is generated by calling: pymc.Matplot.gof_plot(x_sim, x, name='x') A second approach for evaluating goodness of fit using samples from the posterior predictive distribution involves 7.3. Goodness of Fit 61 PyMC Documentation, Release 2.3.5 the use of a statistical criterion. For example, the Bayesian p-value [Gelman1996] uses a discrepancy measure that quantifies the difference between data (observed or simulated) and the expected value, conditional on some model. One such discrepancy measure is the Freeman-Tukey statistic [Brooks2000]: ∑︁ √ √ 𝐷(𝑥|𝜃) = ( 𝑥𝑗 − 𝑒𝑗 )2 , 𝑗 where the 𝑥𝑗 are data and 𝑒𝑗 are the corresponding expected values, based on the model. Model fit is assessed by comparing the discrepancies from observed data to those from simulated data. On average, we expect the difference between them to be zero; hence, the Bayesian p value is simply the proportion of simulated discrepancies that are larger than their corresponding observed discrepancies: 𝑝 = 𝑃 𝑟[𝐷(𝑥sim |𝜃) > 𝐷(𝑥obs |𝜃)] If 𝑝 is very large (e.g. > 0.975) or very small (e.g. < 0.025) this implies that the model is not consistent with the data, and thus is evidence of lack of fit. Graphically, data and simulated discrepancies plotted together should be clustered along a 45 degree line passing through the origin, as shown in Figure 7.7. Fig. 7.7: Plot of deviates of observed and simulated data from expected values. The cluster of points symmetrically about the 45 degree line (and the reported p-value) suggests acceptable fit for the modeled parameter. The discrepancy function in the diagnostics package can be used to generate discrepancy statistics from arrays of data, simulated values, and expected values: D = pymc.discrepancy(x, x_sim, x_exp) For a dataset of size 𝑛 and an MCMC chain of length 𝑟, this implies that x is size (n,), x_sim is size (r,n) and x_exp is either size (r,) or (r,n). A call to this function returns two arrays of discrepancy values (simulated and observed), which can be passed to the discrepancy_plot function in the Matplot module to generate a scatter plot, and if desired, a p value: 62 Chapter 7. Model checking and diagnostics PyMC Documentation, Release 2.3.5 pymc.Matplot.discrepancy_plot(D, name='D', report_p=True) Additional optional arguments for discrepancy_plot are identical to other PyMC plotting functions. 7.3. Goodness of Fit 63 PyMC Documentation, Release 2.3.5 64 Chapter 7. Model checking and diagnostics CHAPTER 8 Extending PyMC PyMC tries to make standard things easy, but keep unusual things possible. Its openness, combined with Python’s flexibility, invite extensions from using new step methods to exotic stochastic processes (see the Gaussian process module). This chapter briefly reviews the ways PyMC is designed to be extended. 8.1 Nonstandard Stochastics The simplest way to create a Stochastic object with a nonstandard distribution is to use the medium or long decorator syntax. See Chapter Building models. If you want to create many stochastics with the same nonstandard distribution, the decorator syntax can become cumbersome. An actual subclass of Stochastic can be created using the class factory stochastic_from_dist. This function takes the following arguments: • The name of the new class, • A logp function, • A random function, • The NumPy datatype of the new class (for continuous distributions, this should be float; for discrete distributions, int; for variables valued as non-numerical objects, object), • A flag indicating whether the resulting class represents a vector-valued variable. The necessary parent labels are read from the logp function, and a docstring for the new class is automatically generated. Instances of the new class can be created in one line. Full subclasses of Stochastic may be necessary to provide nonstandard behaviors (see gp.GP). 8.2 User-defined step methods The StepMethod class is meant to be subclassed. There are an enormous number of MCMC step methods in the literature, whereas PyMC provides only about half a dozen. Most user-defined step methods will be either MetropolisHastings or Gibbs step methods, and these should subclass Metropolis or Gibbs respectively. More unusual step methods should subclass StepMethod directly. 8.2.1 Example: an asymmetric Metropolis step Consider the probability model in examples/custom_step.py: 65 PyMC Documentation, Release 2.3.5 mu = pymc.Normal('mu',0,.01, value=0) tau = pymc.Exponential('tau',.01, value=1) cutoff = pymc.Exponential('cutoff',1, value=1.3) D = pymc.Truncnorm('D',mu,tau,-np.inf,cutoff,value=data,observed=True) The stochastic variable cutoff cannot be smaller than the largest element of 𝐷, otherwise 𝐷‘s density would be zero. The standard Metropolis step method can handle this case without problems; it will propose illegal values occasionally, but these will be rejected. Suppose we want to handle cutoff with a smarter step method that doesn’t propose illegal values. Specifically, we want to use the nonsymmetric proposal distribution: 𝑥𝑝 |𝑥 ∼ Truncnorm(𝑥, 𝜎, max(𝐷), ∞). We can implement this Metropolis-Hastings algorithm with the following step method class: class TruncatedMetropolis(pymc.Metropolis): def __init__(self, stochastic, low_bound, up_bound, *args, **kwargs): self.low_bound = low_bound self.up_bound = up_bound pymc.Metropolis.__init__(self, stochastic, *args, **kwargs) def propose(self): tau = 1./(self.adaptive_scale_factor * self.proposal_sd)**2 self.stochastic.value = \ pymc.rtruncnorm(self.stochastic.value, tau, self.low_bound, self.up_bound) def hastings_factor(self): tau = 1./(self.adaptive_scale_factor * self.proposal_sd)**2 cur_val = self.stochastic.value last_val = self.stochastic.last_value lp_for = pymc.truncnorm_like(cur_val, last_val, tau, self.low_bound, self.up_bound) lp_bak = pymc.truncnorm_like(last_val, cur_val, tau, self.low_bound, self.up_bound) if self.verbose > 1: print self._id + ': Hastings factor %f'%(lp_bak - lp_for) return lp_bak - lp_for The propose method sets the step method’s stochastic’s value to a new value, drawn from a truncated normal distribution. The precision of this distribution is computed from two factors: self.proposal_sd, which can be set with an input argument to Metropolis, and self.adaptive_scale_factor. Metropolis step methods’ default tuning behavior is to reduce adaptive_scale_factor if the acceptance rate is too low, and to increase adaptive_scale_factor if it is too high. By incorporating adaptive_scale_factor into the proposal standard deviation, we avoid having to write our own tuning infrastructure. If we don’t want the proposal to tune, we don’t have to use adaptive_scale_factor. The hastings_factor method adjusts for the asymmetric proposal distribution [Gelman2004]. It computes the log of the quotient of the ‘backward’ density and the ‘forward’ density. For symmetric proposal distributions, this quotient is 1, so its log is zero. Having created our custom step method, we need to tell MCMC instances to use it to handle the variable cutoff. This is done in custom_step.py with the following line: M.use_step_method(TruncatedMetropolis, cutoff, D.value.max(), np.inf) This call causes 𝑀 to pass the arguments cutoff, D.value.max(), and TruncatedMetropolis object’s __init__ method, and use the object to handle cutoff. 66 np.inf to a Chapter 8. Extending PyMC PyMC Documentation, Release 2.3.5 Its often convenient to get a handle to a custom step method instance directly for debugging purposes. M.step_method_dict[cutoff] returns a list of all the step methods 𝑀 will use to handle cutoff: >>> M.step_method_dict[cutoff] [<custom_step.TruncatedMetropolis object at 0x3c91130>] There may be more than one, and conversely step methods may handle more than one stochastic variable. To see which variables step method 𝑆 is handling, try: >>> S.stochastics set([<pymc.distributions.Exponential 'cutoff' at 0x3cd6b90>]) 8.2.2 General step methods All step methods must implement the following methods: step(): Updates the values of self.stochastics. tune(): Tunes the jumping strategy based on performance so far. A default method is available that increases self.adaptive_scale_factor (see below) when acceptance rate is high, and decreases it when acceptance rate is low. This method should return True if additional tuning will be required later, and False otherwise. competence(s): A class method that examines stochastic variable 𝑠 and returns a value from 0 to 3 expressing the step method’s ability to handle the variable. This method is used by MCMC instances when automatically assigning step methods. Conventions are: 0 I cannot safely handle this variable. 1 I can handle the variable about as well as the standard Metropolis step method. 2 I can do better than Metropolis. 3 I am the best step method you are likely to find for this variable in most cases. For example, if you write a step method that can handle MyStochasticSubclass well, the competence method might look like this: class MyStepMethod(pymc.StepMethod): def __init__(self, stochastic, *args, **kwargs): ... @classmethod def competence(self, stochastic): if isinstance(stochastic, MyStochasticSubclass): return 3 else: return 0 Note that PyMC will not even attempt to assign a step method automatically if its __init__ method cannot be called with a single stochastic instance, that is MyStepMethod(x) is a legal call. The list of step methods that PyMC will consider assigning automatically is called pymc.StepMethodRegistry. current_state(): This method is easiest to explain by showing the code: state = {} for s in self._state: state[s] = getattr(self, s) return state 8.2. User-defined step methods 67 PyMC Documentation, Release 2.3.5 self._state should be a list containing the names of the attributes needed to reproduce the current jumping strategy. If an MCMC object writes its state out to a database, these attributes will be preserved. If an MCMC object restores its state from the database later, the corresponding step method will have these attributes set to their saved values. Step methods should also maintain the following attributes: _id: A string that can identify each step method uniquely (usually something like <class_name>_<stochastic_name>). adaptive_scale_factor: An ‘adaptive scale factor’. This attribute is only needed if the default tune() method is used. _tuning_info: A list of strings giving the names of any tuning parameters. For Metropolis instances, this would be adaptive_scale_factor. This list is used to keep traces of tuning parameters in order to verify ‘diminishing tuning’ [Roberts2007]. All step methods have a property called loglike, which returns the sum of the log-probabilities of the union of the extended children of self.stochastics. This quantity is one term in the log of the Metropolis- Hastings acceptance ratio. The logp_plus_loglike property gives the sum of that and the log-probabilities of self.stochastics. 8.2.3 Metropolis-Hastings step methods A Metropolis-Hastings step method only needs to implement the following methods, which are called by Metropolis.step(): reject(): Usually just def reject(self): self.rejected += 1 [s.value = s.last_value for s in self.stochastics] propose(): Sets the values of all self.stochastics to new, proposed values. This method may use the adaptive_scale_factor attribute to take advantage of the standard tuning scheme. Metropolis-Hastings step methods may also override the tune and competence methods. Metropolis-Hastings step methods with asymmetric jumping distributions may implement a method called hastings_factor(), which returns the log of the ratio of the ‘reverse’ and ‘forward’ proposal probabilities. Note that no accept() method is needed or used. By convention, Metropolis-Hastings step methods use attributes called accepted and rejected to log their performance. 8.2.4 Gibbs step methods Gibbs step methods handle conjugate submodels. These models usually have two components: the ‘parent’ and the ‘children’. For example, a gamma-distributed variable serving as the precision of several normally-distributed variables is a conjugate submodel; the gamma variable is the parent and the normal variables are the children. This section describes PyMC’s current scheme for Gibbs step methods, several of which are in a semi-working state in the sandbox directory. It is meant to be as generic as possible to minimize code duplication, but it is admittedly complicated. Feel free to subclass StepMethod directly when writing Gibbs step methods if you prefer. Gibbs step methods that subclass PyMC’s Gibbs should define the following class attributes: 68 Chapter 8. Extending PyMC PyMC Documentation, Release 2.3.5 child_class: The class of the children in the submodels the step method can handle. parent_class: The class of the parent. parent_label: The label the children would apply to the parent in a conjugate submodel. In the gamma-normal example, this would be tau. linear_OK: A flag indicating whether the children can use linear combinations involving the parent as their actual parent without destroying the conjugacy. A subclass of Gibbs that defines these attributes only needs to implement a propose() method, which will be called by Gibbs.step(). The resulting step method will be able to handle both conjugate and ‘non-conjugate’ cases. The conjugate case corresponds to an actual conjugate submodel. In the non-conjugate case all the children are of the required class, but the parent is not. In this case the parent’s value is proposed from the likelihood and accepted based on its prior. The acceptance rate in the non-conjugate case will be less than one. The inherited class method Gibbs.competence will determine the new step method’s ability to handle a variable 𝑥 by checking whether: • all 𝑥‘s children are of class child_class, and either apply parent_label to 𝑥 directly or (if linear_OK=True) to a LinearCombination object (chapter Building models), one of whose parents contains 𝑥. • 𝑥 is of class parent_class If both conditions are met, pymc.conjugate_Gibbs_competence will be returned. If only the first is met, pymc.nonconjugate_Gibbs_competence will be returned. 8.3 New fitting algorithms PyMC provides a convenient platform for non-MCMC fitting algorithms in addition to MCMC. All fitting algorithms should be implemented by subclasses of Model. There are virtually no restrictions on fitting algorithms, but many of Model‘s behaviors may be useful. See Chapter Fitting Models. 8.3.1 Monte Carlo fitting algorithms Unless there is a good reason to do otherwise, Monte Carlo fitting algorithms should be implemented by subclasses of Sampler to take advantage of the interactive sampling feature and database backends. Subclasses using the standard sample() and isample() methods must define one of two methods: draw(): If it is possible to generate an independent sample from the posterior at every iteration, the draw method should do so. The default _loop method can be used in this case. _loop(): If it is not possible to implement a draw() method, but you want to take advantage of the interactive sampling option, you should override _loop(). This method is responsible for generating the posterior samples and calling tally() when it is appropriate to save the model’s state. In addition, _loop should monitor the sampler’s status attribute at every iteration and respond appropriately. The possible values of status are: ’ready’: Ready to sample. 8.3. New fitting algorithms 69 PyMC Documentation, Release 2.3.5 ’running’: Sampling should continue as normal. ’halt’: Sampling should halt as soon as possible. _loop should call the halt() method and return control. _loop can set the status to ’halt’ itself if appropriate (eg the database is full or a KeyboardInterrupt has been caught). ’paused’: Sampling should pause as soon as possible. _loop should return, but should be able to pick up where it left off next time it’s called. Samplers may alternatively want to override the default sample() method. In that case, they should call the tally() method whenever it is appropriate to save the current model state. Like custom _loop() methods, custom sample() methods should handle KeyboardInterrupts and call the halt() method when sampling terminates to finalize the traces. 8.4 A second warning: Don’t update stochastic variables’ values inplace If you’re going to implement a new step method, fitting algorithm or unusual (non-numeric-valued) Stochastic subclass, you should understand the issues related to in-place updates of Stochastic objects’ values. Fitting methods should never update variables’ values in-place for two reasons: • In algorithms that involve accepting and rejecting proposals, the ‘pre-proposal’ value needs to be preserved uncorrupted. It would be possible to make a copy of the pre-proposal value and then allow in-place updates, but in PyMC we have chosen to store the pre-proposal value as Stochastic.last_value and require proposed values to be new objects. In-place updates would corrupt Stochastic.last_value, and this would cause problems. • LazyFunction‘s caching scheme checks variables’ current values against its internal cache by reference. That means if you update a variable’s value in-place, it or its child may miss the update and incorrectly skip recomputing its value or log-probability. However, a Stochastic object’s value can make in-place updates to itself if the updates don’t change its identity. For example, the Stochastic subclass gp.GP is valued as a gp.Realization object. GP realizations represent random functions, which are infinite-dimensional stochastic processes, as literally as possible. The strategy they employ is to ‘self-discover’ on demand: when they are evaluated, they generate the required value conditional on previous evaluations and then make an internal note of it. This is an in-place update, but it is done to provide the same behavior as a single random function whose value everywhere has been determined since it was created. 70 Chapter 8. Extending PyMC CHAPTER 9 Probability distributions PyMC provides a large suite of built-in probability distributions. For each distribution, it provides: • A function that evaluates its log-probability or log-density: normal_like(). • A function that draws random variables: rnormal(). • A function that computes the expectation associated with the distribution: normal_expval(). • A Stochastic subclass generated from the distribution: Normal. This section describes the likelihood functions of these distributions. 9.1 Discrete distributions 9.2 Continuous distributions 9.3 Multivariate discrete distributions 9.4 Multivariate continuous distributions 71 PyMC Documentation, Release 2.3.5 72 Chapter 9. Probability distributions CHAPTER 10 Conclusion MCMC is a surprisingly difficult and bug-prone algorithm to implement by hand. We find PyMC makes it much easier and less stressful. PyMC also makes our work more dynamic; getting hand-coded MCMC’s working used to be so much work that we were reluctant to change anything, but with PyMC changing models is much easier. We welcome new contributors at all levels. If you would like to contribute new code, improve documentation, share your results or provide ideas for new features, please introduce yourself on our mailing list. Our wiki page. also hosts a number of tutorials and examples from users that could give you some ideas. We have taken great care to make the code easy to extend, whether by adding new database backends, step methods or entirely new sampling algorithms. 73 PyMC Documentation, Release 2.3.5 74 Chapter 10. Conclusion CHAPTER 11 Acknowledgements The authors would like to thank several users of PyMC who have been particularly helpful during the development of the 2.0 release. In alphabetical order, these are Mike Conroy, Abraham Flaxman, J. Miguel Marin, Aaron MacNeil, Nick Matsakis, John Salvatier, Andrew Straw and Thomas Wiecki. Anand Patil’s work on PyMC has been supported since 2008 by the Malaria Atlas Project, principally funded by the Wellcome Trust. David Huard’s early work on PyMC was supported by a scholarship from the Natural Sciences and Engineering Research Council of Canada. 75 PyMC Documentation, Release 2.3.5 76 Chapter 11. Acknowledgements CHAPTER 12 Appendix: Markov Chain Monte Carlo 12.1 Monte Carlo Methods in Bayesian Analysis Bayesian analysis often requires integration over multiple dimensions that is intractable both via analytic methods or standard methods of numerical integration. However, it is often possible to compute these integrals by simulating (drawing samples) from posterior distributions. For example, consider the expected value of a random variable x: ∫︁ 𝐸[x] = x𝑓 (x)𝑑x, x = {𝑥1 , ..., 𝑥𝑘 } where 𝑘 (the dimension of vector 𝑥) is perhaps very large. If we can produce a reasonable number of random vectors {xi }, we can use these values to approximate the unknown integral. This process is known as Monte Carlo integration. In general, MC integration allows integrals against probability density functions: ∫︁ 𝐼 = ℎ(x)𝑓 (x)dx to be estimated by finite sums: 𝑛 1 ∑︁ 𝐼ˆ = ℎ(x𝑖 ), 𝑛 𝑖=1 where x𝑖 is a sample from 𝑓 . This estimate is valid and useful because: • By the strong law of large numbers: 𝐼ˆ → 𝐼 with probability 1 • Simulation error can be measured and controlled: 𝑛 ˆ = 𝑉 𝑎𝑟(𝐼) ∑︁ 1 ˆ2 (ℎ(x𝑖 ) − 𝐼) 𝑛(𝑛 − 1) 𝑖=1 (12.1) Why is this relevant to Bayesian analysis? If we replace 𝑓 (x) with a posterior, 𝑓 (𝜃|𝑑) and make ℎ(𝜃) an interesting function of the unknown parameter, the resulting expectation is that of the posterior of ℎ(𝜃): 𝑛 ∫︁ 𝐸[ℎ(𝜃)|𝑑] = 𝑓 (𝜃|𝑑)ℎ(𝜃)𝑑𝜃 ≈ 1 ∑︁ ℎ(𝜃) 𝑛 𝑖=1 (12.2) 77 PyMC Documentation, Release 2.3.5 12.1.1 Rejection Sampling Though Monte Carlo integration allows us to estimate integrals that are unassailable by analysis and standard numerical methods, it relies on the ability to draw samples from the posterior distribution. For known parametric forms, this is not a problem; probability integral transforms or bivariate techniques (e.g Box-Muller method) may be used to obtain samples from uniform pseudo-random variates generated from a computer. Often, however, we cannot readily generate random values from non-standard posteriors. In such instances, we can use rejection sampling to generate samples. Posit a function, 𝑓 (𝑥) which can be evaluated for any value on the support of 𝑥 : 𝑆𝑥 = [𝐴, 𝐵], but may not be integrable or easily sampled from. If we can calculate the maximum value of 𝑓 (𝑥), we can then define a rectangle that is guaranteed to contain all possible values (𝑥, 𝑓 (𝑥)). It is then trivial to generate points over the box and enumerate the values that fall under the curve (Figure Rejection sampling of a bounded form. Area is estimated by the ratio of accepted (open squares) to total points, multiplied by the rectangle area.). f(X) Max f(x) A B X Fig. 12.1: Rejection sampling of a bounded form. Area is estimated by the ratio of accepted (open squares) to total points, multiplied by the rectangle area. Points under curve × box area = lim 𝑛→∞ Points generated ∫︁ 𝐵 𝑓 (𝑥)𝑑𝑥 𝐴 This approach is useful, for example, in estimating the normalizing constant for posterior distributions. 78 Chapter 12. Appendix: Markov Chain Monte Carlo PyMC Documentation, Release 2.3.5 Fig. 12.2: Rejection sampling of an unbounded form using an enveloping distribution. 12.1. Monte Carlo Methods in Bayesian Analysis 79 PyMC Documentation, Release 2.3.5 If 𝑓 (𝑥) has unbounded support (i.e. infinite tails), such as a Gaussian distribution, a bounding box is no longer appropriate. We must specify a majorizing (or, enveloping) function, 𝑔(𝑥), which implies: 𝑔(𝑥) ≥ 𝑓 (𝑥) ∀𝑥 ∈ (−∞, ∞) Having done this, we can now sample 𝑥𝑖 from 𝑔(𝑥) and accept or reject each of these values based upon 𝑓 (𝑥𝑖 ). Specifically, for each draw 𝑥𝑖 , we also draw a uniform random variate 𝑢𝑖 and accept 𝑥𝑖 if 𝑢𝑖 < 𝑓 (𝑥𝑖 )/𝑐𝑔(𝑥𝑖 ), where 𝑐 is a constant (Figure Rejection sampling of an unbounded form using an enveloping distribution.). This approach is made more efficient by choosing an enveloping distribution that is “close” to the target distribution, thus maximizing the number of accepted points. Further improvement is gained by using optimized algorithms such as importance sampling which, as the name implies, samples more frequently from important areas of the distribution. Rejection sampling is usually subject to declining performance as the dimension of the parameter space increases, so it is used less frequently than MCMC for evaluation of posterior distributions [Gamerman1997]. 12.2 Markov Chains A Markov chain is a special type of stochastic process. The standard definition of a stochastic process is an ordered collection of random variables: {𝑋𝑡 : 𝑡 ∈ 𝑇 } where 𝑡 is frequently (but not necessarily) a time index. If we think of 𝑋𝑡 as a state 𝑋 at time 𝑡, and invoke the following dependence condition on each state: 𝑃 𝑟(𝑋𝑡+1 = 𝑥𝑡+1 |𝑋𝑡 = 𝑥𝑡 , 𝑋𝑡−1 = 𝑥𝑡−1 , . . . , 𝑋0 = 𝑥0 ) = 𝑃 𝑟(𝑋𝑡+1 = 𝑥𝑡+1 |𝑋𝑡 = 𝑥𝑡 ) then the stochastic process is known as a Markov chain. This conditioning specifies that the future depends on the current state, but not past states. Thus, the Markov chain wanders about the state space, remembering only where it has just been in the last time step. The collection of transition probabilities is sometimes called a transition matrix when dealing with discrete states, or more generally, a transition kernel. In the context of Markov chain Monte Carlo, it is useful to think of the Markovian property as “mild nonindependence”. MCMC allows us to indirectly generate independent samples from a particular posterior distribution. 12.2.1 Jargon-busting Before we move on, it is important to define some general properties of Markov chains. They are frequently encountered in the MCMC literature, and some will help us decide whether MCMC is producing a useful sample from the posterior. • Homogeneity: A Markov chain is homogeneous at step 𝑡 if the transition probabilities are independent of time 𝑡. • Irreducibility: A Markov chain is irreducible if every state is accessible in one or more steps from any other state. That is, the chain contains no absorbing states. This implies that there is a non-zero probability of eventually reaching state 𝑘 from any other state in the chain. • Recurrence: States which are visited repeatedly are recurrent. If the expected time to return to a particular state is bounded, this is known as positive recurrence, otherwise the recurrent state is null recurrent. Further, a chain is Harris recurrent when it visits all states 𝑋 ∈ 𝑆 infinitely often in the limit as 𝑡 → ∞; this is an important characteristic when dealing with unbounded, continuous state spaces. Whenever a chain ends up in a closed, irreducible set of Harris recurrent states, it stays there forever and visits every state with probability one. • Stationarity: A stationary Markov chain produces the same marginal 80 Chapter 12. Appendix: Markov Chain Monte Carlo PyMC Documentation, Release 2.3.5 distribution when multiplied by the transition kernel. Thus, if 𝑃 is some 𝑛 × 𝑛 transition matrix: 𝜋P = 𝜋 𝑓 𝑜𝑟𝑀 𝑎𝑟𝑘𝑜𝑣𝑐ℎ𝑎𝑖𝑛 : 𝑚𝑎𝑡ℎ : ‘𝜋‘.𝑇 ℎ𝑢𝑠, : 𝑚𝑎𝑡ℎ : ‘𝜋‘𝑖𝑠𝑛𝑜𝑙𝑜𝑛𝑔𝑒𝑟𝑠𝑢𝑏𝑠𝑐𝑟𝑖𝑝𝑡𝑒𝑑, 𝑎𝑛𝑑𝑖𝑠𝑟𝑒𝑓 𝑒𝑟𝑟𝑒𝑑𝑡𝑜𝑎𝑠𝑡ℎ𝑒 * 𝑙𝑖𝑚𝑖𝑡𝑖𝑛𝑔𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖 • Ergodicity: Ergodicity is an emergent property of Markov chains which are irreducible, positive Harris recurrent and aperiodic. Ergodicity is defined as: lim 𝑃 𝑟(𝑛) (𝜃𝑖 → 𝜃𝑗 ) = 𝜋(𝜃) ∀𝜃𝑖 , 𝜃𝑗 ∈ Θ 𝑛→∞ 𝑜𝑟𝑖𝑛𝑤𝑜𝑟𝑑𝑠, 𝑎𝑓 𝑡𝑒𝑟𝑚𝑎𝑛𝑦𝑠𝑡𝑒𝑝𝑠𝑡ℎ𝑒𝑚𝑎𝑟𝑔𝑖𝑛𝑎𝑙𝑑𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛𝑜𝑓 𝑡ℎ𝑒𝑐ℎ𝑎𝑖𝑛𝑖𝑠𝑡ℎ𝑒𝑠𝑎𝑚𝑒𝑎𝑡𝑜𝑛𝑒𝑠𝑡𝑒𝑝𝑎𝑠𝑎𝑡𝑎𝑙𝑙𝑜𝑡ℎ𝑒𝑟𝑠𝑡𝑒𝑝𝑠.𝑇 ℎ𝑖𝑠𝑖𝑚𝑝𝑙𝑖𝑒𝑠𝑡ℎ𝑎 ∫︁ 𝑖+𝑛 1 ∑︁ ℎ(𝜃𝑗 ) ≈ 𝑓 (𝜃)ℎ(𝜃)𝑑𝜃 𝑛 𝑗=𝑖+1 12.3 Why MCMC Works: Reversible Markov Chains Markov chain Monte Carlo simulates a Markov chain for which some function of interest (e.g. the joint distribution of the parameters of some model) is the unique, invariant limiting distribution. An invariant distribution with respect to some Markov chain with transition kernel 𝑃 𝑟(𝑦 | 𝑥) implies that: ∫︁ 𝑃 𝑟(𝑦 | 𝑥)𝜋(𝑥)𝑑𝑥 = 𝜋(𝑦). 𝑥 Invariance is guaranteed for any reversible Markov chain. Consider a Markov chain in reverse sequence: {𝜃(𝑛) , 𝜃(𝑛−1) , ..., 𝜃(0) }. This sequence is still Markovian, because: 𝑃 𝑟(𝜃(𝑘) = 𝑦 | 𝜃(𝑘+1) = 𝑥, 𝜃(𝑘+2) = 𝑥1 , . . .) = 𝑃 𝑟(𝜃(𝑘) = 𝑦 | 𝜃(𝑘+1) = 𝑥) Forward and reverse transition probabilities may be related through Bayes theorem: 𝑃 𝑟(𝜃(𝑘+1) = 𝑥 | 𝜃(𝑘) = 𝑦)𝜋 (𝑘) (𝑦) 𝜋 (𝑘+1) (𝑥) Though not homogeneous in general, 𝜋 becomes homogeneous if Do you ever call the stationary distribution itself homogeneous?: • 𝑛→∞ • 𝜋 (𝑖) = 𝜋 for some 𝑖 < 𝑘 If this chain is homogeneous it is called reversible, because it satisfies the detailed balance equation: 𝜋(𝑥)𝑃 𝑟(𝑦 | 𝑥) = 𝜋(𝑦)𝑃 𝑟(𝑥 | 𝑦) Reversibility is important because it has the effect of balancing movement through the entire state space. When a Markov chain is reversible, 𝜋 is the unique, invariant, stationary distribution of that chain. Hence, if 𝜋 is of interest, we need only find the reversible Markov chain for which 𝜋 is the limiting distribution. This is what MCMC does! 12.3. Why MCMC Works: Reversible Markov Chains 81 PyMC Documentation, Release 2.3.5 12.4 Gibbs Sampling The Gibbs sampler is the simplest and most prevalent MCMC algorithm. If a posterior has 𝑘 parameters to be estimated, we may condition each parameter on current values of the other 𝑘 −1 parameters, and sample from the resultant distributional form (usually easier), and repeat this operation on the other parameters in turn. This procedure generates samples from the posterior distribution. Note that we have now combined Markov chains (conditional independence) and Monte Carlo techniques (estimation by simulation) to yield Markov chain Monte Carlo. Here is a stereotypical Gibbs sampling algorithm: As we can see from the algorithm, each distribution is conditioned on the last iteration of its chain values, constituting a Markov chain as advertised. The Gibbs sampler has all of the important properties outlined in the previous section: it is aperiodic, homogeneous and ergodic. Once the sampler converges, all subsequent samples are from the target distribution. This convergence occurs at a geometric rate. (0) (0) (0) 1. Choose starting values for states (parameters): 𝜃 = [𝜃1 , 𝜃2 , . . . , 𝜃𝑘 ] 2. Initialize counter 𝑗 = 1 3. Draw the following values from each of the 𝑘 conditional distributions: (𝑗) 𝜃1 (𝑗) 𝜃2 (𝑗) 𝜃3 (𝑗−1) (𝑗−1) (𝑗−1) (𝑗−1) ∼ 𝜋(𝜃1 |𝜃2 ∼ ∼ (𝑗) (𝑗−1) (𝑗−1) (𝑗−1) 𝜋(𝜃2 |𝜃1 , 𝜃3 , . . . , 𝜃𝑘−1 , 𝜃𝑘 ) (𝑗) (𝑗) (𝑗−1) (𝑗−1) 𝜋(𝜃3 |𝜃1 , 𝜃2 , . . . , 𝜃𝑘−1 , 𝜃𝑘 ) ∼ 𝜋(𝜃𝑘−1 |𝜃1 , 𝜃2 , . . . , 𝜃𝑘−2 , 𝜃𝑘 ∼ (𝑗) (𝑗) (𝑗) (𝑗) (𝑗) 𝜋(𝜃𝑘 |𝜃1 , 𝜃2 , 𝜃4 , . . . , 𝜃𝑘−2 , 𝜃𝑘−1 ) , 𝜃3 , . . . , 𝜃𝑘−1 , 𝜃𝑘 ) .. . (𝑗) 𝜃𝑘−1 (𝑗) 𝜃𝑘 (𝑗) (𝑗) (𝑗) (𝑗−1) ) 4. Increment 𝑗 and repeat until convergence occurs. 12.5 The Metropolis-Hastings Algorithm The key to success in applying the Gibbs sampler to the estimation of Bayesian posteriors is being able to specify the form of the complete conditionals of 𝜃. In fact, the algorithm cannot be implemented without them. Of course, the posterior conditionals cannot always be neatly specified. In contrast to the Gibbs algorithm, the Metropolis-Hastings algorithm generates candidate state transitions from an alternate distribution, and accepts or rejects each candidate probabilistically. Let us first consider a simple Metropolis-Hastings algorithm for a single parameter, 𝜃. We will use a standard sampling distribution, referred to as the proposal distribution, to produce candidate variables 𝑞𝑡 (𝜃′ |𝜃). That is, the generated value, 𝜃′ , is a possible next value for 𝜃 at step 𝑡 + 1. We also need to be able to calculate the probability of moving back to the original value from the candidate, or 𝑞𝑡 (𝜃|𝜃′ ). These probabilistic ingredients are used to define an acceptance ratio: 𝑎(𝜃′ , 𝜃) = The value of 𝜃(𝑡+1) is then determined by: {︂ ′ 𝜃 (𝑡+1) 𝜃 = 𝜃(𝑡) 82 𝑞𝑡 (𝜃′ |𝜃)𝜋(𝜃′ ) 𝑞𝑡 (𝜃|𝜃′ )𝜋(𝜃) with prob. min(𝑎(𝜃′ , 𝜃), 1) with prob. 1 − min(𝑎(𝜃′ , 𝜃), 1) Chapter 12. Appendix: Markov Chain Monte Carlo PyMC Documentation, Release 2.3.5 This transition kernel implies that movement is not guaranteed at every step. It only occurs if the suggested transition is likely based on the acceptance ratio. A single iteration of the Metropolis-Hastings algorithm proceeds as follows: The original form of the algorithm specified by Metropolis required that 𝑞𝑡 (𝜃′ |𝜃) = 𝑞𝑡 (𝜃|𝜃′ ), which reduces 𝑎(𝜃′ , 𝜃) to 𝜋(𝜃′ )/𝜋(𝜃), but this is not necessary. In either case, the state moves to high-density points in the distribution with high probability, and to low-density points with low probability. After convergence, the Metropolis-Hastings algorithm describes the full target posterior density, so all points are recurrent. 1. Sample 𝜃′ from 𝑞(𝜃′ |𝜃(𝑡) ). 2. Generate a Uniform[0,1] random variate 𝑢. 3. If 𝑎(𝜃′ , 𝜃) > 𝑢 then 𝜃(𝑡+1) = 𝜃′ , otherwise 𝜃(𝑡+1) = 𝜃(𝑡) . 12.5.1 Random-walk Metropolis-Hastings A practical implementation of the Metropolis-Hastings algorithm makes use of a random-walk proposal. Recall that a random walk is a Markov chain that evolves according to: 𝜃(𝑡+1) = 𝜃(𝑡) + 𝜖𝑡 𝜖𝑡 ∼ 𝑓 (𝜑) As applied to the MCMC sampling, the random walk is used as a proposal distribution, whereby dependent proposals are generated according to: 𝑞(𝜃′ |𝜃(𝑡) ) = 𝑓 (𝜃′ − 𝜃(𝑡) ) = 𝜃(𝑡) + 𝜖𝑡 Generally, the density generating 𝜖𝑡 is symmetric about zero, resulting in a symmetric chain. Chain symmetry implies that 𝑞(𝜃′ |𝜃(𝑡) ) = 𝑞(𝜃(𝑡) |𝜃′ ), which reduces the Metropolis-Hastings acceptance ratio to: 𝑎(𝜃′ , 𝜃) = 𝜋(𝜃′ ) 𝜋(𝜃) The choice of the random walk distribution for 𝜖𝑡 is frequently a normal or Student’s 𝑡 density, but it may be any distribution that generates an irreducible proposal chain. An important consideration is the specification of the scale parameter for the random walk error distribution. Large values produce random walk steps that are highly exploratory, but tend to produce proposal values in the tails of the target distribution, potentially resulting in very small acceptance rates. Conversely, small values tend to be accepted more frequently, since they tend to produce proposals close to the current parameter value, but may result in chains that mix very slowly. Some simulation studies suggest optimal acceptance rates in the range of 20-50%. It is often worthwhile to optimize the proposal variance by iteratively adjusting its value, according to observed acceptance rates early in the MCMC simulation [Gamerman1997]. 12.5. The Metropolis-Hastings Algorithm 83 PyMC Documentation, Release 2.3.5 84 Chapter 12. Appendix: Markov Chain Monte Carlo CHAPTER 13 List of References 85 PyMC Documentation, Release 2.3.5 86 Chapter 13. List of References CHAPTER 14 Indices and tables • genindex • modindex • search 87 PyMC Documentation, Release 2.3.5 88 Chapter 14. Indices and tables Bibliography [Akaike1973] 8. Akaike. Information theory as an extension of the maximum likelihood principle. In B.N. Petrov and F. Csaki, editors, Second International Symposium on Information Theory, pages 267–281, Akademiai Kiado, Budapest, 1973. [Bernardo1992] J.M. Bernardo, J. Berger, A.P. Dawid, and J.F.M. Smith, editors. Bayesian Statistics 4. Oxford University Press, Oxford, 1992. [Brooks2000] S.P. Brooks, E.A. Catchpole, and B.J.T. Morgan. Bayesian animal survival estimation. Statistical Science, 15: 357–376, 2000. [Burnham2002] K.P. Burnham and D.R. Anderson. Model Selection and Multi-Model Inference: A Practical, Information-theoretic Approach. Springer, New York, 2002. [Christakos2002] 7. Christakos. On the assimilation of uncertain physical knowledge bases: Bayesian and nonBayesian techniques. 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Cowles and K. Vines. coda: Output Analysis and Diagnostics for MCMC. R package version 0.13-3, 2008. URL http://CRAN.R-project.org/package=coda. [R2010] R Development Core Team. R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, 2010. URL http: //www.R-project.org/. [Raftery1995a] A.E. Raftery and S.M. Lewis. The number of iterations, convergence diagnostics and generic metropolis al- gorithms. In D.J. Spiegelhalter W.R. Gilks and S. Richardson, editors, Practical Markov Chain Monte Carlo. Chapman and Hall, London, U.K., 1995. [Raftery1995b] A.E. Raftery and S.M. Lewis. Gibbsit Version 2.0. 1995. URL http://lib.stat.cmu.edu/ general/gibbsit/. [Roelofs2010] 7. Roelofs, J. loup Gailly, M. Adler. zlib: A Massively Spiffy Yet Delicately Unobtrusive Compression Library. 2010. URL http://www.zlib.net/. [Roberts2007] G.O. Roberts and J.S. Rosenthal. Implementing componentwise Hastings algorithms. 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