C S A S S C C S

C S A S S C C S
CSAS
SCCS
Canadian Science Advisory Secretariat
Secrétariat canadien de consultation scientifique
Proceedings Series 2009/058
Compte rendu 2009/058
CEMAM Workshop on Bayesian
Modelling of Marine
Mammal Population Dynamics
Atelier du CEMAM sur la modélisation
bayésienne de la dynamique des
populations de mammifères marins
27-29 March 2007
Freshwater Institute
501 University Crescent,
Winnipeg, MB, Canada
27-29 mars 2007
Institut des eaux douces
501, University Crescent,
Winnipeg, MB, Canada
Chairperson: Pierre R. Richard
Président : Pierre R. Richard
Fisheries and Oceans / Pêches et Océans
Arctic Aquatic Research Division /
Division de la recherche aquatique dans l’Arctique
501 University Crescent
Winnipeg, Manitoba, R3T 2N6
[email protected]
March 2010
Mars 2010
Foreword
The purpose of these Proceedings is to document the activities and key discussions of the
meeting. The Proceedings include research recommendations, uncertainties, and the rationale
for decisions made by the meeting. Proceedings also document when data, analyses or
interpretations were reviewed and rejected on scientific grounds, including the reason(s) for
rejection. As such, interpretations and opinions presented in this report individually may be
factually incorrect or misleading, but are included to record as faithfully as possible what was
considered at the meeting. No statements are to be taken as reflecting the conclusions of the
meeting unless they are clearly identified as such. Moreover, further review may result in a
change of conclusions where additional information was identified as relevant to the topics
being considered, but not available in the timeframe of the meeting. In the rare case when there
are formal dissenting views, these are also archived as Annexes to the Proceedings.
This workshop was not carried out as a formal Fisheries and Oceans Canada (DFO) Science
Advisory process; however, it is being documented in the Canadian Science Advisory
Secretariat’s (CSAS) Proceedings series as it presents some topics of interest related to the
advisory process.
Avant-propos
Le présent compte rendu a pour but de documenter les principales activités et discussions qui
ont eu lieu au cours de la réunion. Il contient des recommandations sur les recherches à
effectuer, traite des incertitudes et expose les motifs ayant mené à la prise de décisions
pendant la réunion. En outre, il fait état de données, d’analyses ou d’interprétations passées en
revue et rejetées pour des raisons scientifiques, en donnant la raison du rejet. Bien que les
interprétations et les opinions contenus dans le présent rapport puissent être inexacts ou
propres à induire en erreur, ils sont quand même reproduits aussi fidèlement que possible afin
de refléter les échanges tenus au cours de la réunion. Ainsi, aucune partie de ce rapport ne doit
être considéré en tant que reflet des conclusions de la réunion, à moins d’indication précise en
ce sens. De plus, un examen ultérieur de la question pourrait entraîner des changements aux
conclusions, notamment si l’information supplémentaire pertinente, non disponible au moment
de la réunion, est fournie par la suite. Finalement, dans les rares cas où des opinions
divergentes sont exprimées officiellement, celles-ci sont également consignées dans les
annexes du compte rendu.
Le présent atelier n’a pas été tenu dans le cadre officiel du processus des avis scientifiques du
ministère des Pêches et des Océans (MPO). Celui-ci est toutefois documenté dans la série des
comptes rendus du Secrétariat canadien de consultation scientifique (SCCS), car il couvre
certains sujets en lien avec le processus des avis.
CEMAM Workshop on Bayesian
Modelling of Marine
Mammal Population Dynamics
Atelier du CEMAM sur la modélisation
bayésienne de la dynamique des
populations de mammifères marins
27-29 March 2007
Freshwater Institute
501 University Crescent,
Winnipeg, MB, Canada
27-29 mars 2007
Institut des eaux douces
501, University Crescent,
Winnipeg, MB, Canada
Chairperson: Pierre R. Richard
Président : Pierre R. Richard
Fisheries and Oceans / Pêches et Océans
Arctic Aquatic Research Division /
Division de la recherche aquatique dans l’Arctique
501 University Crescent
Winnipeg, Manitoba, R3T 2N6
[email protected]
March 2010
Mars 2010
iii
© Her Majesty the Queen in Right of Canada, 2010
© Sa Majesté la Reine du Chef du Canada, 2010
ISSN 1701-1272 (Printed / Imprimé)
ISSN 1701-1280 (Online / En ligne)
Published and available free from:
Une publication gratuite de :
Fisheries and Oceans Canada / Pêches et Océans Canada
Canadian Science Advisory Secretariat / Secrétariat canadien de consultation scientifique
200, rue Kent Street
Ottawa, Ontario
K1A 0E6
http://www.dfo-mpo.gc.ca/csas/
[email protected]
Correct citation for this publication:
On doit citer cette publication comme suit :
DFO. 2010. CEMAM Workshop on Bayesian Modelling of Marine Mammal Population Dynamics; 27-29 March 2007.
DFO Can. Sci. Advis. Sec. Proceed. Ser. 2009/058.
MPO. 2010. Atelier du CEMAM sur la modélisation bayésienne de la dynamique des populations de mammifères
marins ; 27-29 mars 2007. Secr. can. de consult. sci. du MPO, Compte rendu 2009/058.
iv
SUMMARY
The Center of Expertise in Marine Mammalogy organised a workshop to bring together DFO
and outside experts and review approaches being used to model marine mammal population
dynamics, the current state of development of Bayesian modelling, and to identify challenges
and issues related to this approach in its use on Canadian marine mammals. The CEMAM
Workshop on Bayesian Modelling of Marine Mammal Population Dynamics was held on 27-29
March 2007 at the Freshwater Institute. This report summarizes the conclusions of the
workshop and, in appendices, gives extended abstracts of the workshop presentations and a list
of references on Bayesian modelling.
SOMMAIRE
Le Centre d’expertise sur les mammifères marins (CEMAM) a organisé un atelier regroupant
des experts du MPO et de l’extérieur dans le but d’examiner les approches utilisées pour
modéliser la dynamique des populations de mammifères marins et l’état d’avancement de la
modélisation bayésienne ainsi que pour relever les difficultés et les problèmes reliés à
l’application de cette approche aux mammifères marins au Canada. L’atelier du CEMAM sur la
modélisation bayésienne de la dynamique des populations de mammifères marins a eu lieu du
27 au 29 mars 2007 à l’Institut des eaux douces. Le présent rapport résume les conclusions
formulées pendant l’atelier et contient, en annexe, les résumés complets des exposés donnés
au cours de l’atelier ainsi qu’une liste de références sur la modélisation bayésienne.
v
vi
INTRODUCTION
INTRODUCTION
Over the past decades, the field of ecology
and resource management has seen
increasing use of Bayesian modelling (Dixon
and Ellison 1996; Ellison 2004; Clarke 2005).
This trend is also noticeable in marine
mammal science (Givens et al. 1995; Wade
1999, 2002; Ver Hoef and Frost 2003; Hobbs
et al. 2006). In 2006, the DFO’s Center of
Expertise on Marine Mammalogy (CEMAM)
acknowledged this methodological trend and
recognised the value of these methods in the
work of DFO marine mammalogists, in
particular
with
respect
to
population
monitoring.
Depuis quelques décennies, la modélisation
bayésienne gagne en popularité dans les
domaines de l’écologie et de la gestion des
ressources (Dixon et Ellison, 1996; Ellison,
2004; Clarke, 2005). On remarque également
cette tendance du côté de la science des
mammifères marins (Givens et al., 1995;
Wade, 1999, 2002; Ver Hoef et Frost, 2003;
Hobbs et al., 2006). En 2006, le Centre
d’expertise sur les mammifères marins
(CEMAM) du MPO reconnaissait cette
tendance méthodologique ainsi que la valeur
de ces méthodes dans les travaux des experts
des mammifères marins du MPO, surtout en
ce qui concerne la surveillance des
populations.
Bayesian modelling is an approach that
incorporates the uncertainty associated with
data and prior knowledge on model
parameters used to estimate abundance and
trend of marine mammals. Currently, a number
of DFO projects are underway to develop
population models for marine mammals using
a Bayesian framework (e.g. harp seals, grey
seals, beluga whales). However, these
projects would benefit from discussions with
other experts in the field before they are used
to generate scientific advice. Furthermore,
DFO marine mammalogists not familiar with
these methods would benefit from learning
from other researchers experience using a
Bayesian modelling framework for population
modelling.
La modélisation bayésienne est une approche
qui incorpore l’incertitude associée aux
données et les connaissances a priori
concernant les paramètres des modèles
utilisés pour estimer l’abondance des
mammifères marins et les tendances
connexes. Actuellement, le MPO mène un
certain nombre de projets pour élaborer des
modèles des populations de mammifères
marins en ayant recours à un cadre bayésien
(p. ex. phoques du Groenland, phoques gris,
bélugas). Cependant, les responsables de ces
projets pourraient tirer profit de discussions
avec d’autres experts du domaine avant
d’utiliser les résultats de ces modèles pour
produire des avis scientifiques. En outre, les
experts en mammifères marins du MPO, qui
ne sont pas très familiers avec ces méthodes,
pourraient tirer profit de l’expérience d’autres
scientifiques qui utilisent un cadre de
modélisation bayésien pour modéliser des
populations.
For these reasons, CEMAM organised a
workshop to bring together DFO and outside
experts to review approaches being used, the
current state of development of Bayesian
modelling, and to identify challenges and
issues related to this approach in its use on
Canadian marine mammals.
Le CEMAM a donc organisé un atelier
regroupant des experts du MPO et de
l’extérieur dans le but d’examiner les
approches utilisées pour modéliser la
dynamique des populations de mammifères
marins et l’état d’avancement de la
modélisation bayésienne ainsi que pour
relever les difficultés et les enjeux reliés à
l’application
de
cette
approche
aux
1
mammifères marins au Canada.
WORKSHOP OBJECTIVES AND SUMMARY
OBJECTIFS ET RÉSUMÉ DE L’ATELIER
The CEMAM Workshop on Bayesian
Modelling of Marine Mammal Population
Dynamics was held on 27-29 March 2007 at
the Freshwater Institute. Invited speakers
were: Alf Harbitz (IMR, Norway), Rod Hobbs
(NMML, USA), Michael Kingsley (GNI,
Grønland), Geoff, Evans (DFO Newfoundland)
and Georges Watters (SWFSC, USA). In
addition, ten DFO research scientists and
biologists participated in the workshop and the
meeting was open to observers from the
University of Manitoba (Appendix I).
L’atelier du CEMAM sur la modélisation
bayésienne de la dynamique des populations
de mammifères marins a eu lieu du 27 au
29 mars 2007 à l’Institut des eaux douces. Les
conférenciers invités étaient : Alf Harbitz (IMR,
Norvège), Rod Hobbs (NMML, États-Unis),
Michael Kingsley (GNI, Groenland), Geoff
Evans (MPO, Terre-Neuve) et Georges
Watters (SWFSC, États-Unis). Finalement, dix
scientifiques et biologistes du MPO ont
participé à l’atelier, et des observateurs de
l’Université du Manitoba y ont été invités
(annexe I).
Objectives of the workshop were:
Les objectifs de l’atelier sont les suivants.
-
-
-
-
To review the state of science on
Bayesian modelling of population
dynamics with emphasis on marine
mammal population dynamic models.
To identify statistical issues with model
parameter estimation and produce
guidelines for such modelling (choice
and influence of priors, data load,
convergence
and
autocorrelation
issues, etc.).
To review presentations of a few
working models that have been
implemented in software and problems
in reaching convergence.
To review software or programming
issues and get insights from actual
users/programmers
on
computing
issues.
-
-
-
-
Passer en revue l’état de la science
relative à la modélisation bayésienne
de la dynamique des populations,
notamment en ce qui concerne les
modèles de la dynamique des
populations de mammifères marins.
Relever les problèmes statistiques
associés à l’estimation des paramètres
des modèles et élaborer des lignes
directrices pour cette modélisation
(choix et incidence des données a
priori,
volume
de
données,
convergence
et
problèmes
d’autocorrélation, etc.).
Passer en revue des exposés sur
quelques modèles fonctionnels qui ont
été mis en application dans le logiciel
ainsi que les problèmes concernant
l’atteinte de convergence.
Passer en revue les problèmes de
logiciel ou de programmation et obtenir
des
précisions
de
la
part
d’utilisateurs/programmeurs sur les
problèmes de calcul.
On a invité les conférenciers à ne pas oublier
que l’auditoire est composé d’un éventail de
personnes aux connaissances variées des
méthodes bayésiennes et qu’il convenait
d’utiliser un langage simple et d’expliquer les
termes complexes. On a également invité
The invited speakers were told to keep in mind
that the audience is mixed in its knowledge of
Bayesian methods, so to keep language
simple or explain complex terms. The
audience was encouraged to ask questions of
clarification but to keep them short and defer
2
debate for the discussion period.
l’auditoire à poser de courtes questions pour
obtenir des précisions et d’attendre à la
période de discussion pour en débattre
davantage.
This report summarizes workshop conclusions
on Bayesian modelling generated by the
discussion amongst participants following
presentations and in the final discussion
period. Appendices I and II give the list of
participants and the workshop agenda.
Extended abstracts of the presentations and
discussion specific to the models are provided
in Appendix III. In addition, a list of references
relevant to Bayesian statistics and modelling
(in addition to the ones cited below) has been
compiled in Appendix IV. This annotated list
includes introductory texts for those who are
seeking primers on Bayesian modelling and
some of the software in common use.
Le présent rapport résume les conclusions
formulées au cours de l’atelier sur la
modélisation bayésienne, à la suite des
discussions qui ont suivi les exposés et la
période de discussion finale. Les annexes I et
II présentent la liste des participants et l’ordre
du jour de l’atelier. Des résumés complets des
exposés et des discussions sur les modèles
se trouvent à l’annexe III. En outre, l’annexe IV
donne une liste de références sur la statistique
et la modélisation bayésienne (en plus de
celles énumérées ci-après). Cette liste
annotée comprend les textes d’introduction,
pour ceux qui souhaitent en savoir plus sur les
fondements de la modélisation bayésienne
ainsi que l’information sur le logiciel
couramment utilisé.
WORKSHOP CONCLUSIONS ON
BAYESIAN POPULATION MODELLING
CONCLUSIONS DE L’ATELIER SUR LA
MODÉLISATION BAYÉSIENNE DES
POPULATIONS
Following presentations and discussion of
each model, there was a general discussion
on Bayesian population modelling to distil what
we had learned and address workshop
objectives. Participants chose to order the
discussion in a logical sequence, from model
choice to convergence and other issues:
Après les exposés et les discussions sur
chaque modèle, on a tenu une discussion
générale sur la modélisation bayésienne des
populations pour résumer ce que l’on avait
appris et pour répondre aux objectifs de
l’atelier. Les participants ont décidé de tenir la
discussion dans un ordre logique : du choix du
modèle, en passant par la convergence,
jusqu’à d’autres enjeux.
•
•
•
•
•
•
•
•
•
Choix du modèle
•
•
•
•
•
Choix des données a priori
Model choice
Software and programming issues
Choice of priors
Autocorrelation
Convergence
Sufficiency of data
Other issues
3
Problèmes concernant le logiciel et la
programmation
Autocorrélation
Convergence
Suffisance des données
Autres enjeux
MODEL CHOICE
CHOIX DU MODÈLE
An important consideration in Bayesian
modelling is to have a model that both
represents the population or process and can
answer the question that is posed. Bayesian
methods can give results despite the fact that
the model fails to represent the process
adequately [this is of course also true of nonBayesian models]. It is recommended that a
modeller learn about the problem at hand by
progressing
from
simple
deterministic
modelling to moderate process error modelling
(likelihood methods) before moving on to a full
Bayesian model. This step-by-step approach
gives one a better understanding of the added
complexity of the model as it develops and
improves interpretation of the results of the
final Bayesian analysis. Frequently, several
models are plausible. In such cases, model
choice can be aided by the DIC criterion or by
Bayes factors.
Dans la modélisation bayésienne, il importe
que le modèle à la fois représente la
population ou le processus et réponde à la
question
soulevée.
Les
méthodes
bayésiennes peuvent donner des résultats,
même si le modèle ne représente pas le
processus adéquatement [cela est également
vrai pour les modèles non bayésiens, bien
entendu].
On
recommande
que
le
modélisateur
prenne
connaissance
du
problème à résoudre en passant d’une
modélisation déterministe simple à une
modélisation probabiliste modérée (avec
erreur dans les variables), avant d’en arriver à
un modèle bayésien complet. Cette démarche
progressive facilite la compréhension de la
complexité grandissante du modèle au fur et à
mesure de son développement et améliore
l’interprétation des résultats de l’analyse
bayésienne finale. Souvent, plusieurs modèles
sont plausibles. Dans de tels cas, le choix du
modèle peut être facilité à l’aide du critère
d’information de déviance (DIC, ou Deviance
Information Criterion) ou de facteurs de Bayes.
BAYESIAN SOFTWARE AND
PROGRAMMING ISSUES
PROBLÈMES CONCERNANT LE LOGICIEL
ET LA PROGRAMMATION
The Bayesian modelling method most used by
workshop presenters is the Markov Chain
Monte Carlo (MCMC) Gibbs sampling method.
For that approach to Bayesian modelling,
WinBUGs or OpenBUGS are excellent
programs for people that are new to Bayesian
modelling. They have a serious MCMC Gibbs
sampler with many strong features for
diagnostics and graphics. They are relatively
ease of use and are free. The JAGS program
is another useful free Gibbs sampler. It can
compile models that don’t compile in
WinBUGS or OpenBUGS. It is faster and has
additional functions, such as data simulation
for data sufficency checks. JAGS is not as
easy to use. It has no Windows GUI and it
does not allow cyclic walks as BUGS
programs do.
ADModel Builder does
likelihood and Bayesian (MCMC) modelling
and is extremely fast but it is very difficult to
specify a model [ADMB used to be expensive
La méthode de modélisation bayésienne la
plus utilisée par les présentateurs est la
méthode d’échantillonnage de Gibbs, à savoir
la méthode Monte Carlo par chaîne de
Markov (MCMC). Avec cette démarche de
modélisation bayésienne, WinBUGs ou
OpenBUGS sont d’excellents programmes
pour les néophytes de la modélisation
bayésienne.
Ils
offrent
un
excellent
échantillonneur de Gibbs MCMC doté de
nombreuses caractéristiques fort utiles pour le
diagnostic et les graphiques. Ils sont
relativement faciles à utiliser et sont gratuits.
Le programme JAGS est un autre
échantillonneur de Gibbs utile et gratuit. Il peut
compiler des modèles qui ne se compilent pas
avec WinBUGS ou OpenBUGS. Il est plus
rapide et offre des fonctions supplémentaires,
telles que la simulation de données pour les
vérifications de la suffisance des données. Le
programme JAGS est cependant difficile à
4
to buy but it is now freeware supported by a
non-profit user group].
The mathematical
programming package MatLab also has
functions for MCMC. It requires the user to
learn the language and is very expensive.
utiliser. Il n’a pas d’interface graphique
Windows et ne permet pas les trajets
cycliques
comme
le
permettent
les
programmes BUGS. ADModel Builder, qui
établit la probabilité et effectue la modélisation
bayésienne (MCMC), est extrêmement rapide,
mais il est très difficile de préciser un modèle
[ADMB était dispendieux à l’achat, mais il est
maintenant gratuit grâce au soutien d’un
groupe d’utilisateurs sans but lucratif]. Le
progiciel mathématique MatLab présente
également des fonctions MCMC. Cependant,
l’utilisateur doit apprendre le langage de ce
progiciel, qui est très dispendieux.
The Sampling Importance Resampling (SIR)
approach covered by R. Hobbs is more
efficient than MCMC. It keeps only important
samples and it does not require starting
values. There are no canned programs for
SIR, so it does require knowledge in some
programming language, which will be a
deterrent
to
people
unfamiliar
with
programming.
L’approche
de
l’échantillonnage
avec
rééchantillonnage par importance (Sampling
Importance Resampling, ou SIR) examinée
par R. Hobbs est plus efficace que la méthode
MCMC. Seuls les échantillons importants sont
conservés et il n’exige pas de valeurs de
départ. Comme il n’existe aucun programme
de série pour l’approche SIR, il faut avoir une
certaine connaissance du langage de
programmation, ce qui dissuadera ceux qui
sont peu familiers avec la programmation.
CHOICE OF PRIORS
CHOIX DES DONNÉES A PRIORI
It is preferable to use non-informative priors at
first to learn from results. One can then see if
data within the model data set contain
information that will allow the posteriors to
update the priors of the model parameters.
Caution should be used when considering
priors that are too informative, as there may be
no updating. It is good practice to use priors
that are informative only to the extent
necessary to run the program efficiently.
Scaling of informative priors is also important.
For models to be used in managenent
decision-making it is preferred that informed
priors be supported by objective data as
subjective priors will depend on agreement
amongst experts. Finally, it is important to
find means of quantifying qualitative priors.
Il vaut mieux utiliser des données a priori non
informatives en premier lieu pour apprendre
des résultats. On peut alors voir si les
données de l’ensemble de données du modèle
contiennent l’information qui permettra aux
données a posteriori de mettre à jour les
données a priori des paramètres du modèle. Il
faut faire preuve de circonspection lorsqu’on
utilise des données a priori trop informatives,
car il pourrait ne pas y avoir de mise à jour. La
bonne pratique consiste à utiliser des données
a priori qui sont informatives uniquement dans
la mesure nécessaire pour exécuter le
programme efficacement. Il est également
important de mettre les données a priori
informatives à l’échelle. Pour les modèles
destinés à la prise de décisions de gestion, on
préfère que les données a priori soient
étayées par des données objectives, puisque
les données a priori subjectives dépendront
d’un
accord
entre
des
spécialistes.
Finalement, il est important de trouver des
5
moyens de quantifier les données a priori
qualitatives.
AUTOCORRELATION
AUTOCORRÉLATION
This is not a problem with the SIR algorithm
because each trial is independent. And the
algorithm retains only the important samples.
The Gibbs sampler on the other hand, is a
searching routine and each step in the search
is based on and consequently correlated to the
previous.
In Gibbs sampling programs,
autocorrelation is to be expected in any of
these multi-parameter fitting routines. It can
be dealt with by thinning every n samples.
The choice of a thinning n must be done by
trial and error: setting n and checking for
reduction in auto-correlation. It is best to thin
before sampling; otherwise some WinBUGS
estimates for example may not be correct
Ce problème ne se pose pas avec l’algorithme
SIR puisque chaque essai est indépendant.
L’algorithme ne conserve que les échantillons
importants. L’échantillonneur de Gibbs est
quant à lui une routine de recherche, et
chaque étape de la recherche est fondée sur
l’étape précédente et, par conséquent,
corrélée avec celle-ci. Dans les programmes
d’échantillonnage de Gibbs, on s’attend à ce
qu’il y ait autocorrélation dans toutes ces
routines d’ajustement de paramètres multiples.
Cela peut être obtenu par amincissement à
tous les échantillons n. Le choix d’intervalle
d’amincissement n doit être fait par essais et
erreurs : établissement de n et vérification de
la réduction de l’autocorrélation. Il est
préférable d’amincir avant d’échantillonner,
sinon certaines estimations de WinBUGS, par
exemple, peuvent être incorrectes.
CONVERGENCE
CONVERGENCE
Lack of convergence can be a problem with
data that has little information and weak (uninformative) priors. Convergence can also be
assessed by looking at the sample trace of
estimated parameters. If a trace shows trends
(i.e: searching) rather than a random sampling
behaviour then there is a problem of
convergence . Convergence can be assessed
by doing several runs and comparing them. If
the different runs converge to the same
estimates, then there is good convergence. If
they do not, then there is a convergence
problem.
There
are
tools
in
WinBUGS/OpenBUGS to compare runs.
There are also diagnostic tools in R and S-plus
to check for convergence (CODA, BOA) but
they can be misleading. These tools are
somewhat adhoc empirical takes on
diagnostics, which is strange for Bayesian
analysis.
Another approach is to use Q-Q
plots to compare posteriors and priors.
Reparametization of the model can help
convergence speed. While convergence is not
Le manque de convergence peut survenir
lorsque les données reposent sur peu
d’information et que les données a priori (non
informatives)
sont
peu
solides.
La
convergence peut également être évaluée
avec la trace des échantillons des paramètres
estimés. Si une trace montre des tendances
(i.e.: recherche) plutôt qu’un comportement
d’échantillonnage aléatoire, il y a alors un
problème de convergence. On peut évaluer la
convergence en faisant plusieurs passages du
modèle et en les comparant. Si les divers
passages convergent vers les mêmes
estimations, c’est que la convergence est
bonne. Dans le cas contraire, il y a un
problème
de
convergence.
WinBUGS/OpenBUGS offrent des outils pour
comparer des passages. Il y a également des
outils de diagnostic en R et S-plus pour vérifier
la convergence (CODA, BOA), mais ils
peuvent être trompeurs. Ces outils donnent en
quelque sorte des clichés empiriques
ponctuels pour le diagnostic, qui est
6
a problem with the SIR algorithm, it is
necessary to test the number of trials that are
run for the SIR sampling to determine the
precision of the posteriors. Hobbs indicated
that he typically used between 105 and 106 to
insure 2 significant figure precision in the
percentiles of the posteriors.
surprenant en analyse bayésienne. Une autre
approche consiste à utiliser des graphiques QQ afin de comparer les données a posteriori et
les données a priori. La reparamétrisation du
modèle peut aider à améliorer la vitesse de
convergence. Même si la convergence ne
pose pas de problème avec l’algorithme SIR, il
faut examiner le nombre d’essais qui sont
effectués pour l’échantillonnage SIR afin de
déterminer la précision des données a
posteriori. Hobbs a indiqué qu’il utilisait
d’ordinaire entre 105 et 106 pour obtenir une
précision de deux chiffres significatifs dans les
percentiles des données a posteriori.
DATA SUFFICIENCY
SUFFISANCE DES DONNÉES
The concept that Bayesian modelling is for
data poor situations is debatable. There is
little one can do even with a good dataset, if
there is too much observation error and too
much underlying process error. However, the
advantage of the Bayesian approach is that it
will provide a measure of the sufficiency of the
data by comparison of the posterior to the prior
distributions of the parameters of interest.
More specific to population models, catch data
alone cannot be used to estimate population
size. The model must hinge on the likelihood
of one or more population indices to allow
convergence to an estimate.
L’idée voulant que la modélisation bayésienne
soit utilisée dans les situations où les données
sont rares est discutable. On ne peut faire
grand-chose, même avec un bon ensemble de
données, lorsqu’il y a trop d’erreurs
d’observation et trop d’erreurs de traitement
sous-jacentes. Cependant, l’avantage de
l’approche bayésienne est qu’elle donne une
mesure de la suffisance des données en
comparant les distributions postérieures et
antérieures des paramètres d’intérêt. Dans le
cas des modèles sur les populations, on ne
peut utiliser des données sur les prises seules
pour estimer la taille d’une population. Le
modèle doit s’appuyer sur la probabilité d’un
ou de plusieurs indices de la population pour
permettre la convergence vers une estimation.
OTHER ISSUES MENTIONED
AUTRES ENJEUX MENTIONNÉS
When doing risk analyses based on Bayesian
model outputs, the risk model should assess
the decision criterion from the posterior
distributions of parameter estimates. The use
of a similarly-shaped function to the posterior
distribution, derived from its mean and SD for
example, will reduce the efficiency of the
analyses and introduce more error.
The
analyst should use the parameter posterior
samples to model parameter uncertainty.
With WinBUGs or other similar software, it is
easy to use those samples to project forward.
Lorsqu’on effectue des analyses du risque en
s’appuyant sur les résultats des modèles
bayésiens, le modèle du risque doit évaluer le
critère de décision à partir des distributions a
posteriori des estimations de paramètre.
L’utilisation d’une fonction de forme similaire à
la distribution a posteriori, dérivée de sa
moyenne et de son écart-type, par exemple,
réduira l’efficacité des analyses et amènera
plus d’erreurs. L’analyste doit utiliser les
échantillons a posteriori du paramètre pour
modéliser l’incertitude entourant le paramètre.
Avec WinBUGs ou d’autres logiciels du genre,
il est facile d’utiliser de tels échantillons pour
7
établir des projections.
GENERAL CONCLUSIONS
CONCLUSIONS GÉNÉRALES
Should we be Bayesian?
Devons nous être bayésiens?
Yes, Bayesian methods estimate probability
correctly!
For example, the posterior
distribution of population size gives the true
probability of population estimate being equal
to a particular value, given the prior and the
data. Also, one can’t model process error
using likelihood methods because process
error is nested in sampling error. Bayesian
modelling requires additional efforts but it has
many
rewards
(estimates
of
many
parameters).
Oui, les méthodes bayésiennes donnent une
estimation correcte de la probabilité! Par
exemple, la distribution a posteriori de la taille
de la population donne la probabilité véritable
que l’estimation de la population soit égale à
une valeur particulière, compte tenu des
données a priori et des données. En outre, on
ne peut pas modéliser les erreurs de
traitement en utilisant des méthodes de
vraisemblance du fait que les erreurs de
traitement sont comprises dans l’erreur
d’échantillonnage. La modélisation bayésienne
exige des efforts supplémentaires, mais elle
offre de nombreux avantages (estimation de
nombreux paramètres).
If so, when and how should we use Bayesian Si oui, quand et comment devons-nous utiliser
modelling?
la modélisation bayésienne?
La modélisation bayésienne doit être utilisée
lorsque le modèle est de modéré à complexe
et quand il y a une structure d’erreurs
hiérarchique. WinBUGS ou OpenBUGS
conviennent parfaitement pour les débutants
ainsi que pour de nombreuses applications
plus complexes. Cependant, un certain
nombre de logiciels permettent à ceux qui
cherchent à s’attaquer à des problèmes
complexes (par exemple, JAGS, MatLab,
programmation SIR) d’exprimer toute leur
créativité. Les méthodes bayésiennes doivent
être utilisées lorsqu’on souhaite lier une
décision fondée sur un risque probable
directement lié aux données d’entrée, car elles
permettent l’évaluation simultanée de la valeur
des données dans la prise de décision et du
niveau de risque.
Bayesian modelling should be used when the
model is moderate to complex, and when
there is a hierarchical error structure.
WinBUGS or OpenBUGS are best for
beginners as well as many serious
applications. But there are a number of
software choice which allow for creativity (eg.:
JAGS, MatLab, SIR programming) for those
who seek to tackle complex problems.
Bayesian methods should be used when there
is a desire to link a decision based on
probable risk directly linked to input data
because it will allow simultaneous assessment
of the value of the data in making the decision
and the level of risk.
When should one not bother with Bayesian Dans quelles circonstances doit-on laisser la
modelling?
modélisation bayésienne de côté?
It is not necessary to go through all the trouble
of developing and solving a Bayesian model
when the question and the model are simple
and the adequacy of the data is not in
Il n’est pas nécessaire d’effectuer tout le
branle-bas relatif à l’élaboration et à
l’exécution d’un modèle bayésien lorsque la
question et le modèle sont simples et que
8
question.
The reward is not worth the
substantial effort of Bayesian modeling. Wellestablished empirical methods can do for such
situations. It is worth checking, however, that
the question and model really are simple.
Often they are asserted to be simple for
convenience.
In linear regression, for
example, Bayesian methods can deal with
observations of variable precision with no
extra effort.
l’adéquation des données ne soulève pas de
problème. Les avantages tirés n’équivalent
pas à la valeur de l’effort substantiel requis par
la modélisation bayésienne. Des méthodes
empiriques bien établies peuvent convenir en
de telles situations. Il vaut cependant la peine
de vérifier que la question et le modèle sont
vraiment simples. Souvent, on affirme qu’ils
sont simples pour des raisons de commodité.
Dans la régression linéaire, par exemple, les
méthodes bayésiennes peuvent traiter des
observations de précision variable sans
nécessiter d’effort supplémentaire.
ACKNOWLEDGMENTS
REMERCIEMENTS
Garry Stenson wrote the initial proposal for
this workshop. Garry and Mike Hammill helped
convene the meeting. Unfortunately, on the
week of the workshop, they both had to attend
urgent seal issues and could not participate in
the meeting.
Many thanks to all the
presenters
who
shared
their
unique
experience with their working models with
workshop participants and prepared extended
abstracts for the report. We are grateful to
Jack Lawson for acting as rapporteur on day 2
of the meeting.
Garry Stenson a rédigé la proposition initiale
concernant cet atelier. Garry et Mike Hammill
ont contribué à l’organisation de la réunion.
Malheureusement, la semaine au cours de
laquelle l’atelier a eu lieu, ils ont tous deux dû
s’occuper de questions urgentes concernant le
phoque et n’ont pu prendre part à la réunion.
Un grand merci à tous les présentateurs qui
ont partagé leur expérience unique en matière
de modèles fonctionnels avec les participants
et qui ont préparé des comptes rendus pour le
rapport. Nous tenons également à remercier
Jack Lawson qui a agi en tant que rapporteur
pendant la deuxième journée de la réunion.
9
REFERENCES
RÉFÉRENCES
Clark, J.S. 2005. Why environmental scientists are becoming Bayesians. Ecology Letters,
(2005) 8: 2–14.
Dixon, P. and A.M. Ellison. 1996. Introduction: Ecological Applications of Bayesian Inference.
Ecological Applications: Vol. 6, No. 4, pp. 1034-1035. (see also the group of papers in
this seminal collection / voir également les autres documents de cette collection d’intérêt)
Ellison, A.M. 2004. Bayesian Inference in Ecology. Ecology Letters 7: 509-520.
Givens, G.H., A.E. Raftery and J.E. Zeh. 1995. Assessment of the Bering-Chukchi-Beaufort
Seas stock of bowhead whales using the BALEEN II model in a Bayesian synthesis
framework. Rep. int. Whal. Commn. 45: 345-364.
Hobbs, R.C. K.E.W. Shelden, D.J. Vos, K.T. Goetz, and D.J. Rugh. 2006. Status review and
extinction assessment of Cook Inlet belugas (Delphinapterus leucas). AFSC Processed
Rep. 2006-6, 74 p. Alaska Fish. Sci. Cent., NOAA Natl Mar. Fish. Serv., 7600 Sand Point
Way NE, Seattle WA 98115.
Ver Hoef, J.M., and K.J. Frost. 2003. A Bayesian hierarchical model for monitoring harbour seal
changes in Prince William Sound, Alaska. Environmental and Ecological Statistics 10:
201-219.
Wade. P.R. 1999. A comparison of statistical methods for fitting population models to data. In:
Garner, G.W., S. C. Amstrup, J. L. Laake, B.E.J. Manly, L.L. McDonald, and D.G.
Robertson (eds). Marine Mammal Survey and Assessment Methods. AA. Balkema Publ.
Rotterdam. 287 p.
Wade, P.R. 2002. A Bayesian stock assessment of the Eastern Pacific gray whale using
abundance and harvest data from 1967-1996. J. Cetacean Res. Manage. 4(1): 85-98.
10
APPENDIX I
List of participants:
Invited speakers:
Geoff Evans, Department of Fisheries and Oceans, St.John’s, NL, Canada
([email protected])
Alf Harbitz, Institute of Marine Research (IMR), Tromsø, Norway
([email protected])
Roderick Hobbs, National Marine Mammal Laboratory, NMFS Seattle, WA, USA
([email protected])
Michael Kingsley, Grønlands Naturinstitut, Nuuk, Greenland
([email protected])
George Watters, Southwest Fisheries Science Center, Pacific Grove, CA, USA
([email protected])
Other Participants
Holly Cleator, DFO Winnipeg
Larry Dueck, DFO Winnipeg
Steve Ferguson, DFO Winnipeg
Jean-Francois Gosselin, DFO Mont-Joli
Lei Harris, DFO St.Andrews
Jeff Higdon, University of Manitoba
Jack Lawson, DFO St-John’s
Kathleen Martin, DFO Winnipeg
Lisa Loseto, University of Manitoba
Pierre Richard, DFO Winnipeg
Tim Siferd, DFO Winnipet
Kent Smedbol, DFO St.Andrews
11
APPENDIX II
Workshop agenda
-
27 March 2007:
o
o
Morning:
•
•
Chair remarks
Introduction to Bayesian Modelling: “A tutorial on Bayesian
modelling of harp seals” speaker: Geoff Evans and Working model
1: “Fractions pregnant and population history of Newfoundland
harp seals” speaker: Geoff Evans (Appendix III-B-1).
•
•
Discussion of Model 1
“An introduction to Bayesian modelling” (Appendix III-A) and
Working model 2: “Shrimp estimation models” (Appendix III-B-2)
speaker: Michael Kingsley
Discussion of Model 2
Afternoon:
•
-
28 March 2007:
o
Morning:
•
•
o
Afternoon:
•
•
•
-
Working model 3: “Population dynamics of White and Barents Sea
harp seals” speaker: Alf Harbitz
10:45 Discussion of Model 3
Working model 4: “Population dynamics of (mostly) short-beaked
common dolphins from (at least) three stocks in the eastern
Pacific Ocean” speaker: George Watters (90 min).
Discussion of Model 4
Working model 5: “Bayesian population viability analysis (PVA) for
the Cook Inlet, Alaska beluga population using a SamplingInference-Resampling (SIR) algorithm. Rod Hobbs
29 March 2007:
o
o
Morning:
•
•
General discussion on statistical and programming issues
Workshop conclusions, and reporting plans
•
Administrative matters
Afternoon:
12
Appendix III
A) An introduction to Bayesian modelling
(Michael Kingsley, Grønlands Naturinstitut)
To simplify, there are two statistical views of the world termed ”Frequentist” and ”Bayesian”.
The ’Likelihood’ school falls somewhere in the middle.
What is statistics all about? Statistical induction. We know about Deduction, which is from a
general rule, to make inferences about a particular case
Eg: “Epimenides is a Cretan; all Cretans are liars; Epimenides is a liar.”
Induction is the other way round: from many particular cases, we attempt to infer a general rule.
This is the common experience of human existence (nobody gives us the general rules)
So, we run into a common problem: how many cases do we need, and how reliable is the
general rule that we induce? Statistics is inductive reasoning in a quantitative form; it is the
basis for science - because science is an organised quest for general statements about how the
world works.
We need to weigh the claims of rival general statements.
Classical’ or ’frequentist’ statistics grew out of the school of repeated experiments with cards
and dice, among other things. It pre-supposes that the world is —but our knowledge of it is
uncertain. In other words, general rules exist and are revealed to us (uncertainly) by
experiments. They produce probabilistic statements about the statements we make. For
example, about the mean of a population: if we did this experiment many times; and analysed
the data this way; and formulated our statement thus we could state that the mean lies between
U and L” and we would be right 95% of the time’.
Frequentist statistics leads to: confidence intervals: We construct random intervals, based on
our random experiments/observations. These random intervals may, or may not, contain the
true value but the true value is there, regardless of our intervals. We hope analysis allows us:
- to make statements about our confidence intervals, confidence levels associated with definite
statements, to quantitative tests of hypotheses:
“If these means were equal, we would have observed so large a difference in only 1 of 100
experiments carried out like this one.”
We say these means are different, with confidence in what we say.
- to experimental design
Frequentist thinking lends itself to experimental design because experimental design considers
the hypothetical samples, experimental design has hypothesis testing and error rates in mind,
- to statistical control theory and design of quality control schemes
Example: in 115 births, 60 boys, 55 girls. (unbiased) sex ratio estimate is 60/115 = 0.522
because: if we have a lot of samples of births, and we repeatedly calculate the sex ratio this
way;
13
the mean of our calculated values converges on the true value; confidence if the true sex ratio
was bigger than 60.16%, we would have so few as, or fewer than, 60 boys in only 5% of many
such samples; if the true sex ratio was less than 44.96%, we would have so many as, or more
than, 60 boys in only 5% of many such samples; we have 90% confidence that the true value is
within the interval 45.0 to 60.2%; but to make that statement, we had to consider the probability
of many outcomes that we did not observe. (0 boys, 1, 2, 3, &c.) because we consider many
samples, our calculations include the many possible birth orders; (binomial distribution)
Hypothesis test
If the true average sex ratio was 64%, and we took a sample of 115 births many times, we
would observe 60 boys or fewer in only 0.6% of such samples; we reject the suggestion
(‘hypothesis’) that the sex ratio is 64% (or more) ‘at the 1% level’; we are 99% sure that our
decision is the right one.
Likelihood inference
Likelihoods are relative; only one sample–60 boys, 55 girls; how likely is it? L = k . p^60 . (1p)^55; Maximum Likelihood: the sex ratio (p) is 52.2%; not because it has properties related to
many such samples—but because it maximises the likelihood of this sample.
Confidence?
Likelihood support intervals: find values that reduce the (log) likelihood by a given amount;
(without explanation) a 1.353 l.s.i. corresponds to a 90% confidence interval; for this sample,
such an interval is 44.52% to 59.76%; sex ratios outside this interval reduce—too much— the
likelihood of this sample relative to its greatest possible value; likelihood calculations do not
include the many possible birth orders for the sample; only one sample; therefore only on order;
all orders have the same likelihood; so order is irrelevant;
Bayesian statistics is different
Classical: the world is, but we don’t know about it. The true value is there; we construct a
confidence interval for an estimate; in such a way that it has a specified chance of containing
the true value; probability statements about the interval;
Bayesian: the world isn’t—but it might be. The value has a probability distribution, just like
anything else; probability statements about the true value;
Bayesian statistics accepts probabilistic statements about the world. It produces probability
distributions for the values of parameters that define how the world is. It doesn’t produce
hypothesis tests in quite the same way. How does it do this?
Bayes’ Theorem is a restatement of a simple truism in probability:
p(W&O) = p(W) • p(O|W)
p(O&W) = p(O) • p(W|O);
p(O) • p(W|O) = p(W) • p(O|W)
14
Consequently,
p(W|O) = p(W) • p(O|W) / p(O)
which is Bayes theorem. It essentially states that the probability that the world is like so, given
the observations we (have just) made p(W|O) is equal to the probability that the world was like
that anyway p(W) multiplied by the probability of making those observations if the world was like
that p(W|O) and divided by the probability of making those observations anyway p(O). p(O) is
the probability of making just those observations but that probability is equal to:
∫ p(W) • p(O|W) • dW
It is the sum, over all states of the world, of the probability of the observations given that state of
the world, multiplied by the (prior) probability of that state of the world!
So the bottom line of the Bayesian equation is the integral of the top line, the expression for
p(W|O) is a valid probability distribution (integrates to unity).
In practice, p(O) is treated as a normalising constant; given that ∫ p(W|O) =1,
p(W|O) = p(W) • p(O|W) / ∫ (p(W) • p(O|W)) dW
or more simply:
p(W|O) = k • p(W) • p(O|W)
where:
ƒ
p(W) is the prior (distribution for W). This is a (quantitative) statement of what we knew
about the world, before we made—independent of—the observations we just made;
ƒ
p(O|W) is the likelihood of the observations, conditional on W; and
ƒ
p(W|O) is the posterior distribution for W, where the prior distribution has been updated
by the information contained in the observations.
The major difference between Bayesian and Frequentist statistics is that Bayesian thinking is
symmetrical: the World (W) and the Observations (O) get similar treatment in Bayes’ theorem.
In practice, the world under study (W) consists of some set of variables (parameters); p(W)
(the prior) may (usually does) consist of separate univariate distributions of the individual
variables, or multivariate distributions of sets of variables, or a mixture. p(W|O) (the posterior)
comprises one monstrous multivariate distribution of all the parameters we are considering, but
we typically mostly look at the univariate marginal distributions of individual variables (caution–
correlations!).
When we make observations in daily life, we use them to change our thinking about a lot of
things; Bayesian analysis can use observations to update priors simultaneously on many
variables, even on more variables than we have measurements. So the updating is not
independent, results may be correlated. Caution is appropriate in using/interpreting Bayesian
results.
In Bayesian analysis, priors are both a strength and a weakness. They are a strength because
they represent our previous knowledge. They enable us to use this set of observations to
update it. They are a weakness, because they must be included. They deceive us into
15
importing knowledge we haven’t got. Discussion about priors is a major feature of the
development of Bayesian methods, both in general and in particular.
Why does Bayesian statistics work? Bayesian statistics work because that’s the way our minds
work. We live with a probabilistic world-view (prior). We can accept probabilistic statements
about the state of the world, more easily than confidence statements about the statements we
make. Our minds work in a Bayesian way! It is a symmetrical formulation. Our prior knowledge
and recent observations get equal treatment. There is the possibility of simultaneously
considering many different data sources. There is no need to consider the hypothetical many
repetitions of sampling or experimentation. Results are conditioned on one sample!
So why hasn’t it been used so much before? The philosophical rigour of frequentist statistics is
partly to blame but also the denominator of Bayes theorem ( ∫(p(W) • p(O|W))dW) is terribly
heavy on computing. But, also, it is difficult to formulate a (quantitative) prior distribution that
accurately reflects what we (qualitatively) think we know. The prior is perhaps qualitative. There
is a need to generate priors that are appropriately informative or uninformative depending on the
case at hand. Bayes’ idea of a continual updating probability versus one-shot analyses was not
well received.
What did Reverend Bayes really mean? Perhaps he meant that we have an unstated,
permanent, personal, qualitative prior, that is continually updated with new observations; where
observations come a few at a time. What we do now with Bayesian statistics is assemble all the
observations, construct synthetic priors, run all the observations at once; and then do it all again
next time!! This last approach leaves almost no capability for putting this year’s posterior in as
next year’s prior.
Discussion of presentation
This presentation gave an overview of Bayesian thinking and Bayesian modelling. Discussion
was deferred to the working models in the subsequent presentations and in the conclusions of
the workshop.
B) Bayesian modelling examples
This section summarizes the presentations of five invited speakers on their Bayesian population
models. Four were marine mammal population models and the fifth (Kingsley) is a shrimp model
which could be implemented on a marine mammal population with similar data. An attempt at
doing so on Cumberland Sound belugas was presented but it was not deemed successful for
reasons discussed below. The models are given here in the sequence that they were presented
at the workshop.
B-2: Working model 1: A tutorial introduction to Bayesian modelling of harp seals.
Geoffrey Evans, DFO St John's.
What beliefs are warranted by the evidence?
16
This is a sensible question, related to needs of management; the mathematics called probability
theory helps one address it. I'm not interested in whether this is what probability "really" means
(I'm not sure such a question makes sense.)
In fact the question as posed can't be answered. So the trick is to pick a nearby, proxy
question:
What adjustments to previous beliefs are warranted by the evidence?
This question can be answered, and in many instances we can persuade ourselves that the
answer we get it close enough to the answer to the original question.
Exponential growth of pups
I start with a toy model that nobody believes, to illustrate the way of thinking. We suppose that
any year's population of pups is a constant multiple of the previous year's: the log of pup
numbers is a straight line function of time. We start with a belief that the multiple could equally
be anywhere between its minimum possible value of 0 and its maximum possible value of 1.5
(all pups survive and all females reproduce every year), and that the log of
numbers in the year we choose as starting year could equally be anywhere between 3 and 20
(numbers between 20 and 22,000.) We seek an adjusted belief for what these two parameters
could be, based on a few pup surveys.
It's easy to work out the relative probability of two pairs of parameter values, just be dividing
one instance of Bayes rule by another. All the work in Bayesian analysis comes when we seek
absolute probability densities, which means working out the integral normalizing constant in the
denominator. The method most often used is MCMC, which can conveniently be regarded as
magic for now, which lets us draw random samples from the adjusted (posterior) probability
distribution for parameter values. There is specialized software for doing this
without having to understand too much; I use JAGS.
First, let's address the issue of whether answering the question "What adjustments to beliefs
are warranted?" is close to answering the question "What beliefs are warranted?" Figure 1
shows the posterior distribution for the parameters. This distribution occupies only a tiny
fraction of the prior space (which is indicated by the range of the plot axes). The posterior
range of the multiplier has to be magnified to be visible at all. So it's easy to believe that any
"uninformative" or "vague" prior would lead to about the same posterior.
17
Notice also that the estimates of the two parameters are strongly correlated, which makes
sense but also makes trouble for the Gibbs sampler. One possible strategy is to do PCA on a
preliminary MCMC run and choose as auxiliary random variables distances along the principal
axes. This improves the chance of taking uncorrelated random samples from the posterior, and
in practice greatly improves the chance of convergence.
As well as posterior distributions for the parameters, we also have them for any function of the
parameters we care to compute. In particular we can examine our beliefs about the population
histories. In Figure 2 the red circles show the pup surveys and the red vertical lines show their
standard errors. The black lines do not represent possible trajectories (the red and green lines
do that) but envelopes of trajectories. The following is the most important thing I am going to
say. Concentrate not on the lines but on the spaces between them. There are 7 lines, making
6 intervals of equal probability 0.16. This is supposed to be the probability that the true value
lies within the interval. (The thing that confidence intervals don't mean but people always want
to interpret them as if they did.) If you are asked to decide which interval the true value lies in,
you can do no better than rolling a die, with a 5-to-1 chance that you will be wrong. Prudent
management will take that fact into account. Moreover there is one chance in 25 that the true
value lies outside any of the intervals.
18
The small blue bars at each end are how the ends of the black lines change when the 2004 pup
survey is taken into account. It reduces our uncertainty about 2004 pup numbers, and also
about 1952 pup numbers.
Vague priors tend to be vague about things they notice, and dogmatic about things they don't.
For example the model so far is dogmatic that the multiplier is constant over time. There are
hints in the pup surveys that this may not be true, that the rate of increase has itself been
increasing. Straight mathematics would strongly prefer the increasing-multiplier model. But we
don't: it offends our prior beliefs about the population in the 1950s. 1956 had a pup catch of 360
thousand, the following year half of that. This casts doubt on the pup numbers back then, and
also on the constant multiplier idea. Suggests there is scope for a better model but this is not it.
Preliminary work to get roughly independent auxiliary variables turns out to be
crucial for getting the 3-parameter model to converge.
So far what we have done looks very little different from classical statistics, except that we end
up making statements about the probability that a certain fact about the real world is true. It's
almost "How to be Bayesian if you must." Though one benefit for all the extra work: we never
assumed that all the measurement errors were equal, and in fact supplied a data set including
the standard deviations for each survey. So we went beyond the simplest linear regression
theory at no extra cost beyond the cost of being Bayesian.
19
Fractions of females pregnant
Now we get into some of the reasons why we might want to be Bayesian. One thing that might
change with time is the fraction of females pregnant; and this is what we have the most data on.
There is evidence of year-to-year changes and also of systematic changes; there are years of
little or no data. We seek a description that will allow us to use a year's observations when they
are abundant, and otherwise use observations of the same seals, in the same bit of ocean, in
nearby years. Figure 3 shows fractions pregnant for females from age 4 (magenta; at time of
giving birth; 3 at time of observation) to 8+ (red). The area of a circle is proportional to the
number of females examined. This leads to a hierarchical model, when the true fraction
pregnant in a year is a random variable drawn from some hyperdistribution, and then the
observed number pregnant in a sample is a binomial random variable with the given true
probability. We look in detail at age 8+ females.
We can reject immediately the hypothesis that the true fraction pregnant is the same in every
year. The red symbols in Figure 4 are for a model with a vague prior distribution of mean 0.5 for
each year separately. The black symbols are for a hierarchical model; the dashed vertical lines
show the interquartile range of the posterior distribution.
20
Assuming that no year has information about any other year has, naturally, the best fit to data
(lowest deviance), but has the largest number of parameters to be estimated and contains no
information about unobserved years. How do we decide whether to prefer a model that fits
better but has more parameters? Deviance Information Criterion (DIC), based on the estimated
number of model parameters which is the difference between the average deviance and the
deviance of the average prediction. (Deviance is a funny term: perfect fit has nonzero
deviance.) I can't motivate it. It has been shown to work well in some situations and not in
others. It tends to be used because we need to believe in something. A pure hierarchical model
has a better DIC, but for unobserved years it pays no attention to evidence of a trend over time.
21
We have no parametric model for what a trend might be, and so in Figure 5 we use a moving
average model for the hyperparameters alpha and beta of the Beta hyperdistribution. A pure
MA model (black lines; the dashed lines are the interquartile range; similar to what Brian Healey
used but with a different smoothing procedure) has worse DIC than pure hierarchical. A
hierarchical MA model (red symbols), where the MA procedure is used to get the
hyperparameters, has by a small margin the best DIC. This is not in itself a reason to select it,
but it also accords better with our belief that there is a trend and (especially for making
projections of future pregnancies) we will get more accurate answers by taking it into account.
It has the appealing property of relaxing to the overall average of all the data many years away
from all observed years.
Full ballistic population model
The term ballistic is used to indicate that each cohort is launched with its own pup numbers and
then proceeds deterministically with the given natural mortality rate and catch removals. What
new issues arise in the full model?
1) There are several ages each with its own fraction pregnant. But all ages give birth to
identical pups and therefore, because we don't actually determine the age of a seal giving birth,
there is no way to recover this information later. There could be 5000 more pups born to age 6
mothers and 5000 less to age 7, and we would never know. Thus the pregnancy fractions of
separate ages are irredeemably confounded and we might as well estimate only the age 8+
fraction and adjust the others to have the same number of standard deviations from their
expected values. Pragmatically, this also hugely reduces the number of parameters to be
estimated and makes the program run much faster.
22
2) Pups have a higher natural mortality rate than adults. As a first step we assume pup natural
survival is the square of adult survival over a year. In principle the power could be estimated, in
practice it is confounded and in any case makes no different predictions about things we care
about like seal numbers now.
3) Initial conditions now are a vector of 8 nonpup numbers, not a single starting number (pups
are computed from pregnancy fractions.) In the absence of nonpup surveys at any time (age
distribution of catch is the closest we have) it makes sense to require that the starting age
distribution is close to a stable age structure, including pups, and close to equilibrium. But we
can't require that it be exactly that because we know the pup catch varies hugely between years
and this makes the subsequent age structure far from stable. So, technically, we have another
prior distribution for parameter values that make the starting population not too far from stable.
It is a very indirect, implicit prior, which needs some tricks to define it satisfactorily and then
specify it in JAGS. Figures 6 and 7 show the envelopes for pups and nonpups.
23
4) There are new questions we can ask, like the probability that the seal population has been
declining between 1999 and 2004. The ratio of the two numbers is just another function of
parameters whose marginal distribution we can sample.
5) One can also ask about the posterior distribution for fraction pregnant and whether it differs
from the prior. It turns out it can in years when there are pup surveys (marked in green in
Figure 8), and also at the beginning where stability was not defined very nicely. Some badly
behaved exploratory runs converged to totally unbelievable population histories, and then
compensated by choosing wild outliers for the fraction pregnant in survey years.
24
Deviance calculations estimate that there are 2.6 model parameters, barely more than the 2 of
the model with exponential growth of pups. This means that, despite the posterior adjustments
of pregnancy fractions in the years of pup surveys, there are few effective parameters
associated with the freedom to choose pregnancy.
Discussion
Alf: how did you decide on the smoothing window?
Geoff: after a bit of play, it was fixed at 10 years.
Alf: how can you do a GOF test to the model
George: assess the predictive capacity at the higher level of hierarchy. DIC at the lower level of
hierarchy. It is like cross-validation. Eg: new recruits per streams > low level; regional
recruits is high level
Pierre: how does one project forward?
Geoff: With a pure hierarchical model, as soon as run out of data, you are back to projecting the
average.With a running average, projections would be tied to the recent history. Now the full
seal production model: age-structured model is simplified by assumed that pregnancies at all
ages are perfectly correlated. May not be true but will not know anyway. Survival rates:
strong belief that pups have low survival more than older ones but cannot get at it. Used
informative priors in pregnancy rates. Also used informative priors to start off with a stable
age distribution. Strong influence of data on posterior. Nonpup numbers smoothed because
…. (missed the point made CHECK). Funny that precision greater than latest data close to
latest survey
George: This problem arises in all sorts of age-structured models
Alf: it has to do with basic model. In some cases, will give exponential growth.
Kent: are filling pregnancy rates from priors when no data, which are narrower.
25
Steve: Taylor’s law higher variance with higher mean
Geoff: Comment on Bayesian modelling: advantage: having gone to all this trouble to build this
posterior, we also have random sample of all sorts of other parameters!!! Work real hard at
first but have a lot of bonuses. Fig 8 of 8: black is prior median, red is posterior median; big
differences where there is a pup count. We could argue that pregnant does not result in pup
Rob: why are results inconsistent in differences between red and black; why is different.
Mike H: Is this backward extrapolation?
Geoff: No, it is not a Peter Shelton model. Where go next: should use the age distribution of
the catch. Let’s you get at things like stable age distribution. We could get more out of it than
are now.
Pierre: What software was used?
Geoff: I first used WinBUGS, then switched to JAGS (which runs in Linux) and stayed with it
because it runs models where WinBUGS froze. Also, the author of JAGS is very good to
respond to questions and make changes to his software. The syntax is a bit different from
WinBUGS.
George: I really like the fact that JAGS has a way to simulate data and fit model to data; that is
the best way to determine how much data is needed, i.e., to answer the question of data
sufficiency. JAGS does not allow you to do cyclic walks, you can in WinBUGS. All are useful
in their own way (JAGS, WinBUGS, OpenBUGS) because they offer different options.
Rod: if you put same model in all three, do you get same answer?
George: that is why JAGS was invented, to look at this question.
Geoff: I had correlation issues. JAGS is faster. Also JAGS code has been fixed.
Alf: for convergence, best to let it converge a long time vs using many chains
Geoff: it is easy to see if there is a lack of convergence. JAGS produces only one chain, you
can do two runs separately and compare them. I ran models several times and checked if
the trace showed no trend and answers were qualitatively indistinguishable over a few runs.
Demonstration that there is convergence is difficult; non-convergence is easy
George: diagnostic statistics exist but looking at traces is better. The characteristics of the
wiggle trace that shows convergence is that it looks random, not trendy. MCMC is not
supposed to be a random walk. You can also look at the posterior distribution as the MCMC
goes on. You can see it smooth out. But it is feasible to have a multi-modal distribution. Can
you get convergence if there is a multi-modal distribution
Geoff: You would have trouble since the MCMC will update one mode at a time.
George: CODA and BOA have frequentist takes on Bayesian analysis.
Alf: when I run non-Bayesian model by likelihood and mean square, they often don’t converge
and when I simulate I found I can get crazy values of these parameters. If I get crazy results,
then the model must be wrong. Problem arises when poor data and many parameters to
estimate
George: All models are bad. All make assumptions that are untenable to varying degrees
because they simplify biological processes. That influences the capacity to fit the data.
There is no single approach to get you around this (in either Bayesian or empirical methods).
Geoff: This is the question of which has priority the parameter or the data.
George: you can have 20 years of data but did no perturbation of the population. In that case,
you are forced to use an informative prior.
Geoff: if you insist on getting a definitive answer.
Alf: one problem is that all these parameters are changing over time. We should spend time on
making better models. I am afraid of jumping to Bayesian when have problems.
George: That is a totally fair comment. The Bayesian approach has a suite of problems like
others. And some are similar.
26
B-2 Working model 2: A Working Model: The Stock-Production Model in use for
Assessments of the West Greenland Stock of Northern Shrimp. Carsten Hvingel and
Michael Kingsley (presenter), Grønland Natur Institut.
A model of the dynamics of the West Greenland stock of the Northern Shrimp Pandalus borealis
has been built and taken into use for the assessment of the stock and the formulation of
management advice.
The model is not age-structured, and production is related to stock size by a Schaefer
(quadratic) form1. Each year, catch and predation by cod are removed from the stock and
production is added; a lognormal process error is also included in proceeding from the stock in
one year to the stock in the next. The model uses 5 input data series: catches (1955–date),
standardised CPUE (1976–date), a survey index of biomass (1988–date), cod (predator)
biomass (1955–date), and predation-rate estimates (1989–1992).
Predation is related to the biomasses of predator and prey by a Holling type III function,
depending on 2 parameters, with a lognormal error term. The 4-year series of predation-rate
estimates contributes to the estimation of the parameters of the predation relationship, but the
model is not dependent on these estimates. Both CPUE and survey are considered indices of
stock size, each with its own scaling factor (catchability) and lognormal error. Reported catches
are considered absolute in size and error-free, partly because the objective is to provide advice
on the management of reported catches, and are the only data series that controls the scale of
stock size.
Because of the difficulty of getting absolute estimates of biomass, the working variable for stock
size in the model is its ratio to the Maximum Sustainable Yield Level ( Bmsy). Parameters
affecting the process are therefore: error variances associated with the stock-dynamic process,
CPUE, survey, predation in general, and predation from the 4-year series; catchabilities for the
CPUE and the survey; 2 parameters defining the predation relationship; production rate at
MSYL; and the MSY itself.
The model is built as a Bayesian model formulated for the WinBUGS platform. Non-informative
priors are used, except for the maximum predation rate at high prey stock size (given an
informative Normal prior with truncated tails based on experimental feeding studies) and
carrying capacity (given a slightly informative log-Normal prior with infinite tails in order to
smooth convergence), for both of which there is little information in the data. It runs smoothly
and gives repeatable results, with significant updating of the priors. The modelled stock
trajectory tracks both the biomass indices, and error variances are acceptably small.
1
The model was originally formulated as a Pella-Tomlinson model, allowing a skewed relationship between
production and stock size. The resulting stock-production curve was very close to Schaefer, and the simpler
Schaefer model ran so much faster that the model was simplified to that form.
27
Mean
MSY
215
K
1300
Survey.catch
8.654
CPU.catch
72.48
P.50
4.346
Omax
3
cv.CPUE
2378
cv.Survey
96370
cv.Process
14370
cv.Predation 567400
cv.Grunwald 203200
P[1]
0.9703
Bmsy
650
Zmsy
0.8965
Priors
SD
Median
249.9
99.66
1691
790.9
18.88 0.3153
175.8 0.9993
11.85 0.3965
0.3166
3
283100 0.1518
4.56E+07 0.3731
3.83E+06 0.2631
2.03E+08 0.4787
6.40E+07
3.354
0.1964
0.951
845.7
395.5
2.094 0.2521
Mean
157.1
2181
0.41
1.061
5.651
2.999
0.03846
0.1591
0.1014
0.494
0.8792
0.9172
1091
0.1888
Posteriors
SD
Median Skewness
52.24
145.2 0.68
3310
1515 0.60
0.1894 0.3978 0.19
0.4105 0.9781 0.61
4.047
4.813 0.62
0.3122
2.999 0.00
0.01484 0.03582 0.53
0.0288 0.1558 0.34
0.02621 0.1021 -0.08
0.2943 0.4623 0.32
0.55 0.7623 0.64
0.1933 0.8931 0.37
1655
757.4 0.60
0.06094
0.19 -0.06
Table: Updating of priors in the W. Greenland assessment model for Pandalus borealis
160
140
Catch (Kt)
120
100
80
60
40
20
0
28
90
2000
70
60
1200
50
40
800
30
20
Cod biomass
10
Predation
0
1950
0
1960
1970
1980
600
2000
2010
2.0
500
Survey biomass (Kt)
1990
1.6
400
1.2
300
0.8
200
100
0.4
Survey biomass estimate
Est. std. CPUE (1976=1)
400
Est. predation (Kt)
Est. cod biomass (Kt)
80
1600
Estimated CPUE
0
0.0
Figure. Data series input to the W. Greenland shrimp model: catches, survey and CPUE
indices of bomass, and biomass of predator with a short series of estimated predation values.
The smooth working of the model is probably due to wide ranges in all the data series, good
agreement between the two biomass index series, and a stock history that is consistent with all
the data series: ‘the story hangs together’.
In formulating management advice, the model time series are extended into the future so that all
the information on the joint distribution of the determining parameters is used in predicting the
consequences of management actions. While other parameter estimates—such as error
variances—are used in evaluating model fit, the principal stock-dynamic parameters that enter
into formulating advice are the MSY and the most recent estimates of the relative biomass level.
However, advice is most strongly affected by the predicted probabilities that given catch levels
will take stock levels below, or total mortality above, precautionary limits. The model branches
in the future so that it can simultaneously forecast outcomes for several catch levels, and
several possible trajectories for the predator stock size, several years into the future.
29
Discussion
Alf: If you ask one expert to give a prior probability distribution, he will, and then you ask
another expert and she will give another distribution. Probability is defined subjectively even
if they both meant the same thing.
Michael K: One could ask people to give min, max and median and then one would fit a beta
distribution. If you did that to many people, you could get at the prior distribution of the
parameter.
Alf: if you get enough response, that becomes data that can be analysed; Bayes can disappear.
Michael K: That is a useful dataset for local knowledge of population qualitative statements.
Alf: It is wise to make subjective statements before you get out and get data. So it is a priori.
Michael K: This shrimp model actually works. Initially we used a Pella-Tomlinsen which was
slow to run. We now use a Shaefer or logistic model. Relative estimate of biomass (P =
B/Bmsy) are also easier than estimating biomass directly. Catches were assumed error-free.
CPUE and survey biomass were unscaled indices. Used WinBUGS. Zmsy was stock
productivity at MSY. Omax was the max predation rate. We had convergence to the same
results from different starting values. Priors are updated significantly by likelihood. Why did it
work so well? The range of data on cod stock change is very large. The catch change 6 KT
to 130 KT. Survey and CPUE also varied widely (by factor of 2). The story hangs together:
there few shrimp when cod is high and shrimp increased when cod collapsed. The interquartile range of the posterior fit the survey data and CPUE quite well.
Pierre: The number of parameters estimated was greater than the question originally posed.
Michael K: Don’t extract distribution parameters and then model them. Instead, simply extend
the modelled period beyond the data. You get “free” estimates that way. This is not simply
for efficiency. It is because you cannot easily extract the complexity of the posterior
distributions and the joint multivariate distribution between variables. You push it for all its
worth and get as much as you want. If you use distribution parameters and then
approximate the distribution by a model, then you add uncertainty. Use a step function to
calculate the risk probability.
Jack – How does one model for climate change?
Michael K– There is a big shopping list of things to model there. Global warming may increase
cod.
George – estimating q for surveys is good – marine mammal surveys try hard to get it at one;
have you tried estimating q. I treat my catches as priors. Marine mammal people don’t have
the catch data; inspection of fish catch is thorough.
Tim: shrimp loss rate is not recorded
George: what distribution was used for K?
Michael: I used a lognormal; a uniform would cause WinBUGS to stop.
.
30
B-3: Working model 3: Population dynamics of White and Barents Sea harp seals.
Alf Harbitz, Institute of Marine Research (IMR), Tromsø
Some basics on frequentist-likelihood modelling
Let D be data (observations), a parameter (vector) and f(D| ) the likelihood. In frequentist
analysis the parameters are considered as constants, and it is only the data D that are treated
as stochastic variables.
The maximum likelihood estimator, ML, for , is defined as the value for that maximizes the
likelihood, i.e., the value for that maximizes the probability of having observed the
observations D. This is an appealing criterion, and it can be shown that ML has nice
asymptotic properties such as unbiasedness. In data poor situations, however, ML may be
rather biased.
To find the ML in practice, it is often more convenient to maximize the log-likelihood,
L = log(f(D| )). In complex models it is not a trivial task to find ML, and it is wise to see how
robust the maximizing algorithm is to the choice of initial parameter values.
Once the ML is found, the properties of the estimator can be estimated by letting the ML play
the role as the true -value, and simulate data samples, Dsim, from the model f(Dsim| = ML)
with the same sample size as D. In each run Mlsim is calculated based on Dsim, and statistics
like bias, standard deviation, correlation among estimators, confidence intervals and confidence
regions can be found. In addition one can calculate the same statistics on functions of the
parameters, e.g. a 95% confidence interval for the predicted abundance the next year in a
population dynamics model. In addition, model validation can be performed by comparing the
properties of the residuals based on D with the properties of the residuals from the simulations.
The success of a frequentist-likelihood approach depends strongly on an appropriate model and
a sufficiently large sample of appropriate data. If this is not the case, ML will often reveal this by
providing “crazy” values, e.g. a negative instantaneous mortality rate.
Some basics on Bayesian modelling
Let as before D denote data, a parameter and f(D| ) the likelihood. In Bayesian analysis,
however, the parameter is allowed to be stochastic. All inference about is based on the
posterior distribution of the parameter conditional on data, f( |D):
(1)
f (θ | D ) =
f ( D | θ) ⋅ π(θ)
∫ f ( D | θ) ⋅ π(θ) d θ
=
f ( D | θ) ⋅ π(θ)
f ( D)
θ
where ( ) is the prior describing (subjective) a priori information of in terms of a probability
distribution. Note that the denominator in eq.(1), f(D), does not depend on and thus acts as a
proportionality constant in the posterior. The calculation of f(D) easily becomes prohibitive in
complex models with many parameters.
The Bayesian approach has an intuitive appeal because it allows taking into account prior
(expert) knowledge of a parameter, and it weights data versus subjective knowledge according
to what source is most informative. Note however, that in contrast to the frequentistic approach,
the probability concept no longer is defined precisely. Two experts with the same opinion may
quantify their knowledge in terms of priors differently. In addition, a possible bias of a Bayesian
estimator cannot be estimated.
31
To define a Bayesian point estimator, B, the cost function C( ) of the error = B must be
defined B is then the value of that minimizes the Bayes risk E(C) with regard to the posterior.
For a square cost function, B is the mean of the posterior, while B is equal to the median for
C( ) = | |. For a uniform cost function B is the mode in the posterior. In this case B is found
by maximizing the numerator in eq.(1), which technically is identical to finding a ML estimator.
The last decades the application of Bayesian modelling has been increasingly popular due to
the introduction of Markov Chain Monte Carlo (MCMC) simulations, which circumvents the
problem of calculating the difficult denominator, f(D). MCMC gives a recipe of how one can
simulate samples that converge in distribution to the posterior distribution f( |D) without the
need to calculate f(D). In these simulations the posterior distribution of any function of the
parameters (e.g. a future abundance prediction) is obtained as well. And one may construct
credibility regions from the estimated posterior, which is the Bayesian analogue to confidence
intervals.
One fundamental tool in MCMC to produce Markov chains of samples from f( |D) is the
Metropolis-Hastings algorithm:
(2)
⎛ f (θi +1 | D) ⋅ q (θi | θi +1 ) ⎞
α(θi , θi +1 ) = min ⎜1,
⎟
⎝ f (θi | D) ⋅ q (θi +1 | θi ) ⎠
where is the probability of accepting i+1 as a new value of in the chain, and i+1 is
generated by the sampler, q( . that the common factor f(D) for f( i+1|D) in the numerator and
f( i|D) in the denumerator disappear, so the ratio in the algorithm is easily calculated.
Remarkably, the chain will then converge in distribution to f( |D) for any choice of the sampler
q( . A great challenge, however, is to find a sampler with optimal convergence properties in
terms of efficiency. When the distribution of any parameter conditional on all the others is
known, the Gibbs sampler can be applied. In this case = 1, so the sampling score can be said
to be 100%. In WinBugs the sampler is found automatically, and Gibbs sampler is applied when
possible.
Besides finding an appropriate sampler, a major challenge in MCMC is to decide when the
chain has converged. A complicating factor is that in general the samples in the chain are autocorrelated, even in the case when the sampler provide independent samples of the parameters.
32
One convergence diagnostics tool is to run several chains with different starting values to span
the parameter domain. If the between-chain variance become sufficiently small compared to the
within-chain variance, this is a good indication of convergence.
On the priors
Another fundamental challenge in Bayesian analysis is to assess the sensitivity of the results,
e.g. the prediction of next year abundance in a population dynamics model, with regard to the
choice of priors. This is particularly related to so-called non-informative priors for which there
are no subjective knowledge. Intuitively one would think that it is just to choose a flat prior for .
But since one does not have any information of , one has no information of e.g. log( ) either,
so the prior for log( ) should be flat as well. This, however, is not consistent, a flat prior on is
not consistent with a flat prior on log( ).
There are some general guidance rules, though, for how to choose a non-informative prior. For
example, it is recommended to apply a flat prior for pure location parameters, like in the
normal distribution N( , ), and to apply a prior proportional to 1/ for a pure scale parameter,
. These recommendations are deduced from sound invariance principles. A problem is that the
integral of these priors are infinite, so these priors become so-called improper. Another
challenge in complex models is that it is not always trivial to identify if a parameter is a location
or a scale parameter.
A very particular prior is the Jeffrey’s prior, which is insensitive to parameter transformations in
the sense that it has a unique expression in terms of the Fisher information matrix.
Assume a population dynamics model contains 10 non-informative priors where one is uncertain
what the appropriate priors should be. Say that one is interested to examine how sensitive the
prediction of next year’s abundance is with regard to the choice of non-informative priors, and
two different candidates for each parameter is chosen. This gives 210 ≈ 1000 different
combinations. If the model takes one hour to run, the sensitivity test takes about 40 days. Thus
efficiency is an important task, where the application of e.g. automatic differentiation can be a
very useful investment.
The harp seal population dynamics model
We apply a simplified two-age model to illustrate the challenges involved in a frequentistlikelihood approach as well as a Bayesian approach.
Data
Catch data from 1875, only Russian and Norwegian catches
Pup abundance estimates with cv’s 1998, 2000, 2000, 2002, 2003
Pup abundance indices 1968, 1970, 1973, 1976, 1980, 1985, 1988, 1991
Reproductive data: 1288 females 1962-1993
Age distribution of whelping females: 373 (1980), 401 (1988)
Variables:
N1,…,Nn
N01,…,N0n
= abundance of 1+ animals from years 1 to n
= abundance of pups from years 1 to n
Parameters:
K
M0
= ”carrying capacity”, N1+ abundance in year 1
= mortality rate for pups
33
M0
M
m
F
f
p1,…,pA
A
= exp(M0) = survival rate for pups
= mortality rate for 1+ animals
= exp(M) = survival rate for 1+ animals
= birth rate
= female reproduction rate
= pregnancy rate at age 1 to A
= number of ages
Model:
N1 = K
N 01 = Km ⋅ ( f / 2)
N i = ( N i −1 − Ci −1 ) ⋅ m + ( N 0i −1 − C 0i −1 ) ⋅ m0 , i = 2,..., n
N 0i = f / 2 ⋅ ( N i −1 − Ci −1 ) ⋅ m, i = 2,..., n
α ≡ ( f / 2)m0 = (1 − m) / m (equilibrium)
A−1
⎛
⎞
f = ( F / m ) ⋅ ⎜ (1 − m ) ∑ pa m a −1 + p A m A−1 ⎟
a =1
⎝
⎠
Note that, except for the equilibrium assumption at year 1, the model above is exact if we allow
all parameters to vary in time. If we knew the values of all parameters for all years, the model
would exactly fit the true abundance in time. At the same time, the number of parameters in this
case would be much larger than the number of abundance variables, the latter being twice the
number of time steps (years) because it is a two-age model. Thus an infinite set of parameter
values would exactly fit the true abundances.
In practice we are left with a limited sample size of pup abundance estimates (5) within a narrow
time window (1998-2003), 8 relative pup abundance estimates in the period 1969-1991, and a
long time series of catches (from 1875). For the parameters there are very sparse prior
knowledge other than the fact that no parameter can be negative, and that f and the p’s cannot
exceed one.
There are several ways to approach this typical data poor situation. My own philosophy is to
start with as simplified models (few parameters) as possible to see how far these can be
stretched, before more complex models are applied. Even simple models are hard to fully
understand, a fact that in my opinion is far too often ignored.
In the model development I find a tight cooperation between the statistician and the biologist to
be of uttermost importance. The Canadian sea mammal scientist Gary Stenson expressed the
importance of such a cooperation very good during a visit to Tromsø recently, something like:
“The real potential of scientific progress evolves when the statistician becomes curious about
biology and the biologist becomes curious about statistics in their cooperation”
For the harp seal population it is a common opinion that parameters like instantaneous mortality
rates may depend on abundance densities as well as on environmental factors. As a simple
example I modelled the influence of density on M0 as follows:
M 0,t
⎛N
⎞
= M 00 ⋅ ⎜ 1+ ,t −1 ⎟
⎝ K ⎠
3
34
In this model the natural pup mortality increases rather dramatically as the 1+ abundance
approaches the carrying capacity, K. An interesting feature of this model is that it provided much
better fit to the absolute pup estimates and the pup index data than other (Bayesian) multi-age
models with more parameters where the density regulation was put on other parameters, as
shown in the figure below. This does not mean that one model is better than the other, but it
illustrates the flexibility inherent in even simple models.
Bayesian model with more parameters than in right figure
Model with 4 parameters, multiply vertical scale with 100000.
Without the density regulation I have big problems in estimating ML, e.g. when I try to estimate
the 3 parameters K, M0, and f, and in some runs that work I may get non-biological results.
When the same exercise is done applying MCMC with “biological” priors, apparently appropriate
posteriors are produced. In my opinion this illustrates that one should really be cautious in
applying Bayesian modelling in this situation: The MCMC technology works as a black box
producing apparently reasonable results due to the priors, which easily hide the fact that the
data model (the likelihood) in the first place is odd. Another experience with the frequentist-
35
likelihood approach is that several of the parameters covariate very strongly, which is an
important information that will be obscured by applying independent priors.
Is the Bayesian approach worthwhile?
Pros: Bayes in general is particularly useful when there exist prior knowledge of parameters that
can not be estimated from data. MCMC offers large flexibility and enables complicated models
with many parameters. MCMC is an applicable tool that can be used by non-statisticians
Contras:
In data poor situations with problems of getting appropriate results by non-Bayesian approaches
due to e.g. inappropriate models, apparently reasonable results can be obtained by Bayesian
analysis, obscuring/camouflaging model failure. In data poor situations an important task is to
apply a range of different models, reflecting different biological aspects that the biologist finds
appropriate. A tool like MCMC is technically complicated in terms of e.g. interpretation, and
might easily turn the focus too much away from the biology.
Recommendations:
Much effort should be put in developing appropriate data models in close cooperation between
the statistician and the biologist, beginning with simple few-parameter models and a frequentistlikelihood approach to learn properly the properties of the model. Sensitivity analysis of priors
should be emphasized in Bayesian analysis. Subjective knowledge should be quantified,
experimentally when possible.
Discussion
•
•
•
•
•
•
•
•
•
Is Bayesian probability a “vague” term for non-statisticians? Should we be doing Bayesian
analyses because it is the “right” way, or because it is the easiest way? Different
approaches to a problem will yield a better understanding of the underlying data and
parameter relationships and values.
The point is to get the biologists curious about statistics and the statisticians curious about
biology (Stenson quote from Tromsø).
Alf recommends a variety of approaches be applied to a problem such as traditional
statistics, Bayesian models, sensitivity analyses etc.
Discussion of AD model builder - fast in operation. ADMIT module in MATLAB may be
another package of use.
WinBUGS has an extensive help and example system in the main menu.
Should we discuss finding a standard approach to Bayesian analyses (approaches and
software choices)? E.g., JAGS may compile faster, and allow more complex models, than
WinBUGS.
OpenBUGS seems to give better feedback on model errors than WinBUGS, although either
programme would be a good choice for beginning modellers.
Great care must be taken in parameterization as the different programmes require
parameters input in different ways.
Data poor systems (e.g., bowhead pre-whaling population size) may constrain the
functionality of the Bayesian approach (although this may be true of all statistical
36
•
approaches). The context of the problem will have a big impact on the efficacy of the
various statistical approaches.
SIRS versus MCMC approaches were discussed – some of the decision will be predicated
upon the programming skills of the user. SIRS may be an easier programme to work with, if
you are already a competent programmer; may be easier to understand the SIRS system for
a simple model with relatively few parameters. Hobbs will prepare a description of SIRS to
present.
37
B-4 Working model 4: Population dynamics of (mostly) short-beaked common dolphins
from (at least) three stocks in the eastern Pacific Ocean.
George M. Watters, NMFS SWFC.
Using a model that is currently under construction, I provide working examples of various issues
that arise in using the Bayesian approach to model the population dynamics of marine
mammals, and although the title of my presentation gives emphasis to short-beaked common
dolphins, my presentation is intended to be less about dolphins and more about modeling
issues. I attempt to provide examples that illustrate the modeling process, and, since the work
is still in progress, it is important that results illustrated in my presentation are not interpreted as
final results. I am happy to discuss Bayesian modeling, either in general or for specific
applications, with anyone. I have made my code available to participants of the Workshop, but,
since this is an active area of my own research, I ask that users be discreet with its use and
distribution.
I describe the dynamics of short-beaked common dolphins (Delphinus delphis) with an
“extended” Pella-Tomlinson model that is cast in a state-space framework. These extensions
include:
1.
simultaneous fits to estimates of abundance developed from line-transect surveys in two
regions and for three stocks (the northern, central, and southern stocks),
2.
“movement” parameters which determine the probabilities that animals from the northern
and southern stocks are observed in surveys that do not cover the entire range of these stocks
(but one survey does cover the range of the central stock), and
3.
time-varying carrying capacities for each stock.
I explicitly consider how environmental conditions affect movement by the northern and
southern stocks and the carrying capacities of all three stocks. I also use auxiliary information
collected from dip nets to index carrying capacity for the central stock and fit to a time series of
such indices. I include both observation and process error in the model (this is essentially the
state-space framework). Observation error is associated with the line-transect estimates of
abundance, the environmental indices used to model time-varying carrying capacities, and the
dip net data. Process error is included both in the sub-model that describes time-varying
carrying capacity and in the overall dynamics equation described by the Pella-Tomlinson model.
Posterior distributions for the parameters and other, derived variables of interest are estimated
using the MCMC methods implemented in OpenBUGS, and the graphics provided in my
presentation typify the graphics that can be produced from the OpenBUGS GUI (graphical user
interface).
The remainder of this abstract is organized in subsections that address each of the issues
which the Workshop Convenors identified as topics of specific interest.
Choice of prior distributions
My example model has numerous parameters, and I use a variety of approaches to specify their
prior distributions.
1.
I develop some priors on the basis of information in the literature. For example, results
from Reilly and Barlow (1986) suggest that it is unlikely, but nevertheless possible, that the
maximum rate of population increase at low abundance (r) is about 8% per year. Thus, I use a
“folded normal” prior (by taking the absolute value of a normally distributed random variable) for
38
r and specify a variance for this distribution that allows one to infer that a value of 8% per year is
not very likely.
2.
I develop some priors by using approximations that provide a desired prior on a derived
or implied parameter. For example, it might be desirable to place a uniform prior distribution on
the maximum net productivity level (MNPL, expressed as the ratio of abundance to carrying
capacity). However, in the familiar Pella-Tomlinson framework, there is not an analytical
solution for z (the productivity shape parameter) in terms of MNPL; thus I cannot sample directly
from a prior for MNPL and use algebra to convert these sample values into values of z. There
is, however, an analytical solution for MNPL in terms of z, so I specify a prior distribution for z
that provides an approximately uniform prior on MNPL over the range [0.5, 0.8]. Such an
approximation is provided in my code.
3.
I develop some priors by specifying means, variances, and ranges that provide
biologically sensible results or do not cause numerical problems. For example, in a case where
I compute the natural logarithm of the difference of two parameters with independent marginal
prior distributions, I ensure that each marginal prior is specified in a way that does not allow the
difference to be negative or extremely close to zero, either of which would cause numerical
errors in calculating the logarithm. This type of problem can also be addressed by specifying
appropriate joint priors, but I find that it can be difficult to communicate about joint prior
distributions to some audiences (and even to myself!).
4.
I develop some priors by adopting a suite of “common sense” rules. For example, I
illustrate an S-shaped model that relates the “true” carrying capacities (K) estimated in the
Pella-Tomlinson framework to “observations” of carrying capacity recorded in the dip net data.
This S-shaped model has a parameter that defines an intercept, and I specify that the prior
distribution for this parameter should include the possibility of an intercept near zero because it
seems reasonable that dip net catches might be negligible when the true K is small (of course
this does not have to be the case if there is reason to suspect a positive bias in observations of
K).
5.
I develop some priors on the basis of my own “prior belief.” For example, I use
penalized splines to smooth some environmental data within the assessment model (illustrating
how uncertainty associated with constructing environmental indices can be carried all the way
through an assessment without having to do the smoothing “outside” the model), and I specify a
prior belief about the degree of smoothness for these splines. I implement this belief by
specifying a prior distribution which puts substantial weight on the hypothesis that one source of
variance in the smooth (the variance of the random errors) is about twice as big as another
source of variance (the variance of the smoothing parameters themselves). In another
example, I control the degree to which carrying capacities can vary over time by using a box
constraint to specify the prior belief that K cannot change (either positively or negatively) by
more than 10% per year.
Convergence of MCMC chains and autocorrelation of posterior samples
In my opinion, convergence to a stationary distribution and autocorrelation in the posterior
samples can often be assessed visually. I usually assess convergence to a stationary
distribution by plotting “parameter traces.” I consider whether traces of single chains have
properties like those which represent parameters that are sampled from prior distributions but
are not modified by likelihoods. The key here is to recognize that the job is to assess
convergence to a distribution rather than to a point. Therefore, do not expect parameter traces
39
to converge to straight lines. I also start multiple chains from different locations in the
parameter space and consider whether these chains converge to similar distributions. I learned
to make visual assessments of parameter convergence both by “training” my eyes in exercises
where I viewed parameter traces in tandem with the results of more formalized convergence
diagnostics (e.g., the diagnostics available in the R packages BOA and CODA) and by simply
plotting traces of samples drawn from known distributions. Despite my reliance on visual
diagnostics, formal tests may be a necessary standard in actual stock-assessment situations. It
is important to recognize that an apparent lack of convergence (indicated either visually or by
more formal tests) may actually be a result. For example, if the posterior distribution of a
parameter is bimodal, the trace of that parameter may bounce back and forth between two
separate regions of the parameter space. More generally, an apparent lack of convergence
may usefully indicate that the estimation problem has multiple solutions. I evaluate
autocorrelation in posterior samples using barcharts that plot the value of the autocorrelation
coefficient (the heights of the bars) versus a lag that describes the distance between two
samples from the same MCMC chain. Generally, I hope to see that the heights of the bars
decrease rapidly with increasing lags and that this decrease bottoms out with bar heights that
are close to zero (indicating little autocorrelation) at relatively small lags.
In my experience, difficulties with convergence and autocorrelation are frequent (with this
frequency increasing as models become more complex) but can often be overcome with a few
practical solutions. How a model is parameterized plays a major role in determining both the
rate at which posteriors converge to stationary distributions and the degree to which posterior
samples are autocorrelated. When I am confronted with slow convergence and autocorrelation,
I usually consider how I might reparameterize my model. Centering and standardizing
covariates (e.g., as one might do in the context of a simple linear regression) is often helpful.
Convergence can also be slowed by prior distributions that are overly uninformative, and,
therefore, when I am confronted with slow convergence I often re-evaluate my relevant prior
distributions. If computing time is not important, a brute-force solution to autocorrelation is
simply to produce huge posterior samples (e.g., millions or tens of millions) but then discard
most of these results and keep, say, every 100th value. This process is called “thinning.” Quick
checks of autocorrelation plots like those described above can be used to guide the selection of
a useful thinning interval – i.e., try to pick a thinning interval which corresponds to a lag that has
low autocorrelation for multiple parameters. There are formal diagnostics that can also be used
to help identify a useful thinning interval. The drawback of thinning is, again, that many more
samples are collected than are actually used in making posterior inference (thus it can be
inefficient from a computing standpoint).
Sufficiency of data
In my opinion, the easiest method to assess whether data are sufficient for parameter
estimation is to conduct a Bayesian analysis and compare the posterior distributions to their
respective prior distributions, but the “best” method is probably to simulate data with specific
properties (e.g., sample sizes, CVs, relationships to covariates, etc.) and determine whether
these types of data are in fact sufficient. The easy approach is easy because when posteriors
are different than their priors the data have obviously been sufficiently informative. Comparing
posteriors and priors should not, however, be limited to comparisons of means; it is important to
compare the distributions themselves (e.g., by comparing a range of quantiles, or making qqplots). The more difficult, but preferable, approach may require a substantive investment in
programming and subsequent analysis, but the payoff is gaining an understanding of data
sufficiency more generally, with insight as to what type and quantity of data are “needed,” rather
than knowing whether the data at hand are good enough.
40
Although the simulation approach mentioned above can help indicate the type and quantity of
data that may be sufficient in the future, it does not alleviate the problem of having to deal with
data that are not sufficient for parameter estimation in the present. There are various options
for dealing with this situation (i.e., needing to develop a model, estimate parameters, and
provide advice despite insufficient data), and I attempt to illustrate them in my presentation.
One option is to be forthright about the lack of sufficient data and live with the uncertainty it
brings. For example, the data illustrated in my presentation are not sufficient to estimate the
MNPL (my posterior and prior for this derived parameter are the same), but I am comfortable
with this result precisely because I am uncertain about MNPL and its value plays a critical role
both in determining how a population responds to fishing and, at least in the context of the U.S.
Marine Mammal Protection Act, in assessing the status of marine mammal populations.
Another option is to bring an alternative source of data to bear on the same problem. For
example, in an attempt to estimate how the carrying capacity of the central stock has changed I
use both a series of sea-surface height data (which are the only type of data available for the
northern and southern stocks) and observations of potential dolphin prey organisms collected
during research cruises (which were not available for the other two stocks). Finally, it can be
useful to conduct a sort of Bayesian sensitivity analysis where, in the absence of sufficiently
informative data, informative priors and fixed parameter values are used to build contrasting
scenarios. For example, in my model for D. delphis, I show how estimates of stock status
(based on the ratio of abundance at any point in time to the abundance when a stock is at
MNPL) are sensitive to different priors for parameters in the sub-model relating the “true”
carrying capacity for the central stock to observations collected during research cruises as well
as to different assumptions about the relative amounts of process and observation error in my
model.
Software and programming issues
My presentation here is a simple set of bullet points – it is a bit difficult to combine these points
into a coherent paragraph structure. Many of these points are specific to WinBUGS,
OpenBUGS (which I use most frequently), and JAGS.
•
I have found that programming skills in WinBUGS are transferable to OpenBUGS and
JAGS (and vice versa). The syntax and programming flow in all three packages is similar but
sufficiently different that code developed for one package will usually require some editing
before it can be run in another package.
•
It is important to understand how each modeling package parameterizes its distributions
(e.g., by specifying estimates of variance or estimates of precision).
•
MCMC chains can be thinned at two times during a modeling session within WinBUGS
and OpenBUGS, during sampling or after sampling. Users should be aware that if thinning is
conducted after sampling some of the default graphics will not be developed from the thinned
chain. To ensure that graphics are developed from thinned chains it seems best to thin during
the sampling process, this can also reduce memory requirements and the sizes of output files.
•
I often aim to include various bits of “data processing” within my models (e.g., in my
presentation I illustrate smoothing a series of environmental data within the larger model of
dolphin population dynamics). The advantage of such a thing is that uncertainty associated with
the data processing (e.g., how much to smooth the environmental series) can be integrated into
the assessment and carried through into the results that are relevant to providing management
advice. The disadvantages of this approach include added model complexity and increased
computing time.
41
•
Following from the point above, I also aim to predict as many “values of interest” (e.g.,
metrics like the ratio of abundance at any point in time to the abundance when a stock is at
MNPL and estimates of sustainable mortality like potential biological removals, PBR) within my
models as possible. The advantages here are that, again, uncertainty is integrated into these
values of interest and their posterior distributions are computed as a matter of course, but this
comes at the price of increased computing time.
Discussion
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Geoff: left out the key PRO thing is that calculates the things that conf limits say they do but
they don’t
Alf: yes; but for me it does not solve anything because the Bayesian probability is vague
since includes prior.
Alf: the best thing is to do both empirical and Bayesian.
There are insights to be gained from a progressing form deterministic, sensitivity analysis, to
empirical, to Bayesian (also sensitivity) and insights are gained from all methods
Geoff: ADmodel Builder does all those things at a price
Matlab also had Admit module [ask Alf]
Alf: Gaussian Markov Random Fields GMRF is another branch of stats that is very efficient.
I have seen an example where they replace MCMC. Uses full conditionals
Rod: Discuss standardization Bayesian methods and software
George: not in favour; stifles creativity
Alf: WinBUGS demo
Alf: recommend WinBUGs but if make loops in WinBUGs do not repeat variable in model
code
George: any oddity in WinBUGs is when use lognormal you have to get the log(x) and put it
in dlnorm()
Michael: be careful at all distributiondefinitions; they are different from other software.
George: I use R to look at distribution and then BUGs is totally different
Pierre: sample it to see if correctly specified
George: precision rather than SD
Alf: Compiling is often problematic in WB
Geoff: JAGS will compile models that WB does not
George: cyclic graphs take long to compile
Open BUGs gives more information than WB but still unclear what say
OpenBUGs also has diamonds that can click on and sometimes track through and figure out
WB error messages – scour the WB listserv – few answers but often no answers
As a tool for getting into it, WB is great
Steve: Data poor systems are problematic ; maybe we chose a bad case to start
Initial understanding is that data poor situations can be solved by MCMC but am thinking
differently now.
George: if have limited data, a lot of modelling approaches regardless, all have problems
Michael: if try to do likelihood methods, you run into very slow things; don’t know how to
build into the likelihood method the process error. Easier in WB to build large models.
I checked a model in Excel and got same
The bonus is the extras parameters estimates
George: is context dependent
Huge models in ADModel builder take a lot more work than in WB. It is context-dependent.
Consensus that WB is a good tool to beginners
George: alternative for beginners is to do SIR if competent programmer.
You can invest effort in learning WB or in programming a SIR also in language competent in
42
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SIR is 10 lines in Fortran
MatLab 10-15 lines
Why are people not using SIR?
SIR is more straightforward to understand if small vector size
No search routine involved
If posterior has oddities, like more than one maximum, you will find it easier by SIR rather
than MCMC
Can someone prepare
Geoff: model says that if SSH above normal, K goes up forever
George: true, not totally satisfied by the formulation
A prior belief is that K increases slowly Kt+1/K does not change more than 10% per year.
This debatable. May be something for sensitivity analysis.
Rob: what if there is a sudden change in current, that would be dampened by your
GIVES example of alternative parametization for K that runs faster but worse mixing;
Smoothing depends on how much 1/sd2 is; how smooth or how rough do you want this to
be
Prior weight is that there is more process error than observation error.
Can do all that in one model rather than seperatly
g(K) there is information about inflection point
Sensitivity analysis
[QUESTION]: how to decide what constraints to place on error terms in priors]
Sensitivity shows bias-variance trade-off is gnarly.
RESULTS
WinBUGS graphics are given; not so bad
N/MNPL if below 1 depleted
N as fished / N if no-fishing shows that fishing does little to changes.
Can plug in uncertainty from Bayesian model into PBR calculation
Red line assumes MNPL at 0.5K; Black dots is using the MNPL in model estimates. Black
dots are lower.
Did a second run with reparametized parameters to do PBR
B-5: Model 5: Bayesian population viability analysis (PVA) for the Cook Inlet, Alaska
beluga population using a Sampling-Inference-Resampling (SIR) algorithm.
Rod Hobbs, NMML, NOAA/NMFS, USA
A detailed population model was developed for the Cook Inlet beluga to assess the extinction
risks faced by this small population. The model included immature and mature phases of both
sexes (i.e., age- and sex-structured) and focused on the behaviour of a declining population at
sizes less than 500 belugas. Small population effects were taken into account by examining
survival and fecundity under a range of scenarios that considered demographic stochasticity,
harvest, density dependence, Allee effects, constant mortality effects (e.g., predations), and
unusual mortality events (e.g., catastrophes). Details of the model are available in Hobbs et al.
2006.
The PVA analysis was conducted to produce 3 results of interest:
1) A posterior distribution of the annual growth multiplier for the population.
2) A posterior distribution of population projections to 300 years.
3) A comparison among different types of mortality and fecundity effects in small
populations.
43
A Sampling-Inference-Resampling (SIR) algorithm was used to generate a set of 10,000
values of the annual growth multiplier and associated parameters drawn from the posterior
distribution. The annual growth multiplier is a derived parameter resulting from the choice of
model parameters such as survival and fecundity probabilities. Although different survival
and fecundity values result in the same annual growth multiplier each parameter set will have
a different likelihood. For this analysis 100,000 trial parameter sets are drawn (Sampling) the
model is run and the likelihood is calculated for each trial set (Inference) and a subset is
drawn with replacement from the trial set weighted by the likelihood (Resampling) to develop
the posterior parameter set. The advantage of this method being that this posterior
parameter set then contained all of the necessary information to generate the products of
interest without further sampling.
Data available to the analysis included 13 years of annual abundance estimates and harvest
data for the Cook Inlet population. Life history and population parameters estimated for this
and other beluga populations are available from the literature. Life history parameters of
particular interest for modeling purposes were: survival probability, birth interval, age at first
birth, gestation period, and lactation period.
The model was set up with nearly independent uniform priors for the population
abundance in 1994, N1994 , and the annual growth multiplier, φ0 . Other parameters for density
dependent survival, and fecundity were drawn from priors informed by values from the liturature.
For computational convenience, if N1994 had a likelihood greater than 10-6 × likelihood of N1994
= N 1994 (the abundance point estimate for 1994), the parameters and age structure were
retained as inputs to the Bayesian analysis (parameter values with lower likelihood would have
no influence on the results). Each population was then projected from 1994 to 2005 and
likelihood was calculated as:
Lj =
2005
⎛ Nt, j − N t
∏ T ⎜⎜ N CV ( N
t =1994
⎝
t
⎞
, DF = 10 ⎟
⎟
t)
⎠
(7)
where,
L j is the relative likelihood of the jth population projection;
T ( X , DF = 10 ) is the density of Student’s-t distribution at X with 10 degrees of freedom;
N t , j is the population size of the jth projection in year t; and
N t , CV ( N t ) are the estimated abundance (point estimate) and associated coefficient of
variation in year t.
Projections to 2005 with likelihoods less than 10-10 × the maximum possible likelihood
(i.e., the likelihood if the model Nt was equal to the abundance point estimate in all years) were
discarded as having no contribution to the posterior distribution. A Sampling-ImportanceResampling (SIR) algorithm was followed in which the acceptable parameter sets were
weighted by their relative likelihoods from projections to 2005, and a resample drawn with
replacement to give a posterior distribution of outcomes. Projections to 2305 (300 years into
the future) for this posterior parameter set were done to estimate the probability of decline and
extinction during that period. Model comparisons between the various models were done using
the Bayes factor, calculated as twice the natural logarithm of the ratio of the average likelihoods
of the two resamples:
44
⎡ SIRtot
⎤
⎢ ∑ L j,x
⎥
⎢ j =1
⎥
SIRtot ⎥
BayesFactor ( x, y ) = 2 ln ⎢ SIRtot
⎢
⎥
L j, y
⎢∑
⎥
j =1
⎢⎣
SIRtot ⎥⎦
(8)
where BayesFactor(x,y) is the Bayes factor comparing model x and model y;
ln[] is the natural logarithm of the value in [], Lj,x and Lj,y are the likelihoods of the jth projection of
model x and model y, respectively; and SIRtot is the number of projections in the SIR
subsample.
Where the Bayes factor had absolute value greater than 2 the model with the higher average
likelihood was considered to be the more likely of the two, otherwise the models were of
equivalent likelihood.
All models were compared to the Baseline model ( φ0 = U[0.94, 1.06]) using the Bayes factor.
The three options for modifying the Baseline model, the constant mortality effect (C), the
unusual mortality event (PMe) and the Allee effect (A) were each considered. The time series of
abundance and harvest data covered a sufficient range of population sizes (270-660 belugas) to
compare between the Baseline and the Healthy Population models but not among the remaining
options. Six models with the modifiers for survival and fecundity were considered, three of the
Baseline with the C parameter at 1, 2 or 5 mortalities per year, one of the Baseline with the
Allee parameter at 0.50, one of the Baseline with an unusual mortality event (PMe = 0.05), and
one of the Baseline with the C parameter at 1 and an unusual mortality event (PMe = 0.05 ). Two
additional models were included to test the sensitivity of the parameters: a Baseline and Healthy
Population model that included an unusual mortality event and a C of five mortalities. For each
model, 100,000 trials were projected to 2005 and the likelihood was calculated. Each
population projection was fully defined by 13 parameters, however the parameter of interest was
φ0 , a derived parameter. A sample of 10,000 of these trials, weighted by the likelihoods, was
drawn with replacement for the SIR algorithm resample for further analysis. For all populations
the population size in 1994, 2005, 2105, 2205 and 2305 was retained and for declining
populations the year that the population dropped below 200, 100, 10, and 2 animals was
retained. A population with 1 or 0 individuals was considered extinct.
Reference
Hobbs, R.C. K.E.W. Shelden, D.J. Vos, K.T. Goetz, and D.J. Rugh. 2006. Status review and
extinction assessment of Cook Inlet belugas (Delphinapterus leucas). AFSC Processed
Rep. 2006-6, 74 p. Alaska Fish. Sci. Cent., NOAA Natl Mar. Fish. Serv., 7600 Sand Point
Way NE, Seattle WA 98115.
Discussion
Time did not allow discussion of this model. Discussion of its methods was deferred to the
general discussion on the following day. The results of that discussion are presented in the
main body of the report.
45
Appendix IV
Additional references on Bayesian methods and modelling
Books:
Albert, J. 2007. Bayesian Computation with R. Springer. x + 267 p.
Bolker, B.M. 2008. Ecological Models and Data in R. Princeton University Press. vii + 396 p.
[Excellent text on ecological modelling methods, with many R code examples, including
Bayesian modelling.]
Clark, J.S. 2007. Models for Ecological Data: An introduction. Princeton University Press. xiii +
617 p. [Excellent text on ecological modelling methods, including Bayesian modelling.]
Clark, J.S. 2007. Statistical Computation for Environmental Sciences in R: Lab Manual for
Models for Ecological Data. Princeton University Press. [R code for modelling methods,
including Bayesian modelling.]
Gelman, A., Carlin, J.B., Stern, H.S., and D.B. Rubin. 2004. Bayesian Data Analysis.
Chapman and Hall /CRC Press. 668 p. [Advanced text on Bayesian analysis]
Hacking, I. 2001. Introduction to Probability and Inductive Logic. Cambridge University Press.
302 p. [ A careful discussion of the issues by a philosopher, with a lot of instructive
puzzles (Geoff Evans)]
Hilborn, R. and M. Mangel. 1997. The Ecological Detective: confronting models with data.
Princeton University Press, xvii + 315 p. [Excellent introductory text on likelihood and
Bayesian population modelling methods.]
Jaynes, E.T. 2003. Probability theory: the logic of science. Cambridge University Press,
Cambridge, UK. 758 p. [ text on probability theory; a wonderfully polemic and dogmatic
view (G. Evans)]
King, R., Morgan, B.J.T., Gimenez, O. and S.P. Brooks. 2010. Bayesian Analysis for
Population Ecology. Chapman and Hall/CRC Press. xiii + 442 p. [Excellent text on
Bayesian ecological modelling methods, including population models, with appendices on
R and WinBUGS Bayesian programming.]
McCarthy, M.A. 2007. Bayesian Methods for Ecology. Cambridge University Press. xiii + 296
p. [Excellent text on Bayesian modelling methods, with a primer on WinBUGS use.]
Punt, A. and R. Hilborn. 2001. BAYES-SA - Bayesian Stock Assessment Methods in Fisheries
- User's Manual, http://www.fao.org/DOCREP/005/Y1958E/y1958e00.htm#Contents
[Good primer on various Bayesian Fisheries population modelling methods; downloadable
Excel spreadsheets are used to demonstrate the algorithms. Good reference list!]
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Papers:
(See also references listed above in Introduction)
Innes, S. and R.E.A. Stewart. 2002. Population size and yield of Baffin Bay white whale
(Delphinapterus leucas) stocks. NAMMCO Scientific Publications 4: 225-238.
Wade, P.R., 2002. Bayesian population viability analysis. P. 213-238 In: Beissinger, S.R.,
McCullough, D.R. (Eds.), Population Viability Analysis. University of Chicago Press,
Chicago. 577 p.
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Appendix V
Software links
ADMB Foundation: http://admb-foundation.org/
ADMB Project: http://admb-project.org/
Bayesian Computation with R: http://bayes.bgsu.edu/bcwr/
Bayesian Methods for Ecology: http://arcue.botany.unimelb.edu.au/bayes.html
BOA: http://www.public-health.uiowa.edu/boa
CODA: http://www-fis.iarc.fr/coda/
JAGS: http://www-ice.iarc.fr/~martyn/software/jags/
MatLab: http://www.mathworks.com/
OpenBUGS: http://mathstat.helsinki.fi/openbugs/
PBSadmb (ADMB from R): http://code.google.com/p/pbs-software/
R: http://cran.r-project.org/ and
http://cran.r-project.org/web/views/Bayesian.html
Running WinBugs and OpenBugs from R: http://www.stat.columbia.edu/~gelman/bugsR/
SIR method: No canned software is available but see Hobbs et al (2006), Hilborn and Mangel
(1997) and Punt and Hilborn (2001) and Wade (1999) for examples, spreadsheets or
pseudocode.
WinBUGS: http://www.mrc-bsu.cam.ac.uk/bugs/
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