Embracing uncertainty in fisheriesstock assessment using Bayesian

hierarchical models

Henni Pulkkinen

Academic dissertation

To be presented for public examination with the permission

of the Faculty of Biological and Environmental Sciences

of the University of Helsinki

in Auditorium YB210, Linnanmaa campus, University of

Oulu, on February 27th, 2015, at 12 a.m.

Fisheries and Environmental Management Group

Department of Environmental SciencesFaculty of Biological and Environmental Sciences

University of Helsinkiand

Natural Resources Institute Finland

HELSINKI 2015

Supervisors Prof. Samu MantyniemiFisheries and Environmental Management GroupDepartment of Environmental SciencesUniversity of HelsinkiHelsinki, Finland

Prof. Jukka CoranderBayesian Statistics GroupDepartment of Mathematics and StatisticsUniversity of HelsinkiHelsinki, Finland

Follow-up Prof. Jaakko ErkinaroNatural Resources Institute FinlandOulu, Finland

Prof. Sakari KuikkaFisheries and Environmental Management GroupDepartment of Environmental SciencesUniversity of HelsinkiHelsinki, Finland

Reviewers Ph.D. Carmen FernandezICES Advisory CommitteeInternational Council for the Exploration of the Seas (ICES)Copenhagen, Denmark

Ph.D. Len ThomasCentre for Research into Ecological and EnvironmentalModelling (CREEM)School of Mathematics and StatisticsUniversity of St AndrewsSt Andrews, Scotland, United Kingdom

Examiner Prof. Mikko SillanpaaDepartment of Biology and Department ofMathematical SciencesUniversity of OuluOulu, Finland

Custos Prof. Sakari KuikkaFisheries and Environmental Management GroupDepartment of Environmental SciencesUniversity of HelsinkiHelsinki, Finland

c© Henni PulkkinenISBN 978-951-51-0559-2 (paperback)ISBN 978-951-51-0560-8 (PDF)http://ethesis.helsinki.fiJuvenes PrintOulu 2015

CoverElina PaivaniemiPhoto: Mikko Paartola, vastavalo.fi

Abstract

Overfishing and environmental changes impose high risks on the wellbeing ofthe world’s fish stocks. It is commonly acknowledged that fisheries managementshould be risk averse, following the principles of the precautionary approach, butunfortunately the statistical stock assessment methods often lack the ability toestimate the uncertainties related to their results. Further challenges arise fromthe fact that stocks which are in the most desperate need of a stock assessmentare often data poor and resources to gather new data from them are scarce.

Bayesian statistical inference can be utilized to conduct stock assessments cost-efficiently since these methods provide a formal way to combine information fromvarious sources including databases, literature and expert knowledge. Bayesianinference is essentially a learning process where existing information is combinedinto a prior distribution, which is further updated with the most recent data. Theresult, a posterior distribution, expresses the best available knowledge about thephenomenon, including the related uncertainty. Furthermore, Bayesian hierarchi-cal models enable learning between similar units, for example, stocks of the sameor related species.

This thesis consists of four studies that use Bayesian hierarchical models toimprove knowledge in the fisheries stock assessment. Correlations between bio-logical parameters, arising from different life history strategies among species, areutilized in paper [I] so that a data rich set of length-weight parameters can reducethe uncertainty of length-fecundity parameters for a data poor species. In paper[II], similarities between stocks of Atlantic salmon are used to estimate the stockspecific key parameters of eggs-to-smolts relationship. A predictive distribution ofthis key parameter is also estimated, and could be used as an informative prior ina subsequent study of another Atlantic salmon stock. In addition, a hierarchicalmodel is built to study structural uncertainty, estimating posterior probabilities ofcompeting functional forms that describe the eggs-to-smolts survival at differentjuvenile densities.

The Bayesian approach makes it possible to conduct analyses sequentially, asin the case of Baltic salmon stock assessment reviewed in paper [III]. Sequentialanalysis is useful if a stock assessment is complex and computational power doesnot enable analysis of all observation models at the same time. Thus, the Bayesianmethods permit the creation of complex model frameworks, where incentives forstructural choices arise rather from the biological process, than from the require-ments or limitations of the datasets. The analysis of acoustic survey informationto estimate herring resources at Bothnian Sea (paper [IV]) is another example of acomputationally intensive model that has been built on the basis of available back-ground information. This includes on one hand, the technical knowledge aboutthe survey, and on the other hand, the biological knowledge about the spatialdistribution of the fish.

To my parents, who alone know the length of the journey.

LIST OF ARTICLES INCLUDED

I Pulkkinen H., Mantyniemi S., Kuikka S. and Levontin P.

2011. More knowledge with the same amount of data: ad-vantage of accounting for parameter correlations in hierar-

chical meta-analyses. Marine Ecology Progress Series. 443:29-37.

II Pulkkinen H. and Mantyniemi S. 2013. Maximum survival

of eggs as the key parameter of stock-recruit meta-analysis:accounting for parameter and structural uncertainty. Cana-

dian Journal of Fisheries and Aquatic Sciences. 70: 527-533.

III Kuikka S., Vanhatalo J., Pulkkinen H., Mantyniemi S. andCorander J. 2014. Experiences in Bayesian inference in

Baltic salmon management. Statistical Science. 29: 42-49.

IV Pulkkinen H., Gardmark A., Kaljuste O., Perala T., LiljaJ., Bryhn A. and Mantyniemi S. A Bayesian hierarchical

spatial model for the Baltic International Acoustic Survey(BIAS) and Bothnian Sea herring (Clupea harengus). Sub-

mitted.

The individual contributions of the authors of articles I - IV

I Henni Pulkkinen had the main responsibility in formulating the modelstructure, choosing the data, conducting the analysis and in writing thepaper. Samu Mantyniemi provided the original idea and participated inthe model construction and writing. Sakari Kuikka and Polina Levontinassisted in the literature search and writing.

II Henni Pulkkinen had the main responsibility in formulating the modelstructure, conducting the analysis and in writing the paper. SamuMantyniemi provided the original idea for the paper and assisted inthe model construction and writing.

III Henni Pulkkinen assisted in writing the paper by describing the BalticSalmon stock assessment and provided the graphical illustrations forthe paper. Sakari Kuikka provided the original idea for the paper.Sakari Kuikka and Jarno Vanhatalo had the main responsibility in plan-ning and writing the paper. Samu Mantyniemi and Jukka Coranderparticipated in writing.

IV Henni Pulkkinen had the main responsibility in formulating the modelstructure, conducting the analysis and writing the paper. Olavi Kaljusteprovided the knowledge about the acoustic survey and data analyzedin the paper and assisted in writing. Anna Gardmark and SamuMantyniemi provided the original idea and assisted in writing. SamuMantyniemi also assisted in the model construction. Tommi Peralaassisted in the computational experimentations. Juha Lilja providedtechnical knowledge about the acoustic survey. Andreas Bryhn assistedin writing.

Contents

1 Introduction 2

2 Choices and sources of data 52.1 Data bases and literature . . . . . . . . . . . . . . . . . . . . . 52.2 International Council for the Exploration of the Seas (ICES) . 7

3 Methods 83.1 Bayesian inference . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Hierarchical models . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Structural uncertainty . . . . . . . . . . . . . . . . . . . . . . 123.4 Sequential models . . . . . . . . . . . . . . . . . . . . . . . . . 133.5 Hierarchical spatial models . . . . . . . . . . . . . . . . . . . . 153.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4 Results 164.1 Learning from correlations between biological parameters . . . 164.2 Accounting for model uncertainty in the stock-recruit relation-

ship of Atlantic salmon . . . . . . . . . . . . . . . . . . . . . . 164.3 Updating knowledge about Baltic salmon recruitment with a

sequential analysis . . . . . . . . . . . . . . . . . . . . . . . . 194.4 Introducing uncertainty into the estimates of Baltic Interna-

tional Acoustic Survey . . . . . . . . . . . . . . . . . . . . . . 22

5 Discussion 23

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1 Introduction

The field of fisheries stock assessments is filled with different sources of uncer-tainties. Knowledge about the abundance, reproductive success, migrationpatterns or schooling behavior of fish stocks is at best limited. Many factorsinfluence the status of the stocks, including the availability of food, level ofpredation, environmental factors and human influence from fishing, eutroph-ication or pollution. The information available about the status of stocks,and about the many factors affecting them, is often sparse, biased and con-tradictory. Despite these challenges, there is a need to have firm knowledgeabout the condition of the stocks and to give appropriate management adviceon how they can be harvested sustainably (Hilborn and Walters 1992).

As the task of accounting for uncertainties in a stock assessment can seemoverwhelming, it is often considered easier to concentrate only on the piecesof the puzzle that are best known, instead of dealing with the uncertain andunknown. This can lead into a false feeling of certainty, and in the worstcase for a fish stock, into a situation where a decrease in abundance cannotbe detected before it is too late to prevent the stock from collapsing (Mullonet al. 2005). As fisheries legislation in Europe is expected to follow the pre-cautionary approach (CEC 2009), fisheries policy should be risk averse, ac-knowledging the most important sources of uncertainty in stock assessmentsand expecting those uncertainties to be estimated and taken into account inthe management decisions. Thus, an increase in the estimated uncertaintyshould lead to a more conservative level of exploitation (Fenichel et al. 2008)which, on the other hand, can also result in greater long term gains if theover-fishing of a stock is avoided (Mantyniemi et al. 2009). However, anincrease in the knowledge about a stock does not necessarily mean that theuncertainty decreases: it may even increase when more factors and differingviews are comprehensively accounted for. For example, it may turn out thatthe variability of a biological parameter is greater than what was previouslythought. This may happen especially if the biological realism is improved ina stock assessment, increasing the understanding about the related sourcesof uncertainty (Kuparinen et al. 2012).

Harwood and Stokes (2003) distinguish the epistemic, or measurable, un-certainty into categories of process stochasticity, observation error, modelerror and implementation error. The process stochasticity is considered tocover the natural variation arising from the demography and environmentalvariation affecting the species. Further, Harwood and Stokes (2003) dividethe observation error between the measurement error, including the issues ofdata collection, and the estimation error that reflects the imprecision of thestatistical inference.

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Parameter uncertainty arises from the simple notion that the true valueof a parameter is not known. Process and observation uncertainties, forexample, contribute to the estimated parameter uncertainty. Typically, ifuncertainty is accounted for in a fisheries stock assessment, the focus is onthe parameter uncertainty, rather than on other sources of uncertainty (Hillet al. 2007). For example, uncertainty in the parameters of the stock-recruitrelationship has been widely studied (e.g. Myers and Mertz 1998, Su et al.2004, He et al. 2006). The stock-recruit relationship describes the connectionbetween the spawning stock size and the resulting number of offspring and ithas a big influence on the future projections of the stocks, and, thus, on themanagement advice (Hilborn and Walters, 1992). On the other hand, manyparameters essential for stock assessment, e.g. natural mortality, are stilloften treated as known constants (Gislason et al. 2010), which may result infalse conclusions about the status of the stock.

Model uncertainty, or structural uncertainty, arises from dubiety aboutwhich one of alternative functional forms best describes the phenomenon(Brodziak and Legault 2005, Hoshino et al. 2014). Structural uncertaintyis found to play a large role especially in ecosystem models, where differentspecies can have a wide range of interactions (Hill et al. 2007). This sourceof uncertainty can be accounted for, e.g., by including the competing modelstructures in the same model framework and estimating the probabilities ofdifferent models according to the data (Hoeting et al. 1999). As statisticalmodels are commonly acknowledged as simplifications of the real world, it of-ten is very easy to speculate about the risks that a potential mis-specificationof a model can bring to the management actions. However, it is less com-mon that the optional model structures are rigorously considered, aiming toformally deal with the structural uncertainty.

Implementation uncertainty arises from human behavior, from not know-ing how well the management actions, such as regulations on the total al-lowable catch, are followed in practice. As the fisheries management systemsoften fail in creating the desired behavior among fishermen, the uncertaintyat the human side of the system may even overcome the other sources ofuncertainty (Fulton et al. 2011). Because the complexity of environmentalproblems can include the impact of market forces and personal incentives ofthe resource users, accounting for implementation uncertainty often requiresinterdisciplinary research between societal, economic and biological sciences(Haapasaari et al. 2012).

In order to account for these various sources of uncertainty, a formalmethodological framework is required. Methods of Bayesian inference haveproven beneficial in this task both for fisheries stock assessment and decisionanalysis (e.g. Punt and Hilborn 1997, McAllister and Kirchner 2001, Parma

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2002). Thus, their usage is constantly increasing within the field. Unlikethe standard stock assessment methods (e.g. Virtual Population Analysis,Pope 1972), Bayesian methods make it possible to combine different typesof information such as expert knowledge (Uusitalo et al. 2005), informationfrom other related stocks or species (Jiao et al. 2011), or to sequentiallyanalyze data from different parts of a life cycle and then combine the resultsinto a stock assessment (Michielsens et al. 2008). Furthermore, Bayesianinference makes it also possible to study probabilities of events that have notbeen observed (e.g. McGrayne 2011). This is useful for fisheries science asthe stock collapse is something we wish never to observe but may want tohave an estimate of the risk of such an event occurring.

The focus of this thesis is on providing formal statistical tools for ac-knowledging and reducing parameter and structural uncertainties in fisheriesstock assessment. This thesis consists of four papers, in all of which Bayesianhierarchical models are used for dealing with a specific issue related to a stockassessment. Bayesian hierarchical models are particularly useful for dealingwith uncertainty because they make it possible to transfer information fromthe data rich cases to the data poor, enabling learning between, for example,different individuals, species or geographical areas (Punt et al. 2011). Thus,hierarchical models can either help in evaluating the level of uncertainty in anovel case with sparse data, or in reducing the uncertainty by incorporatingadditional information.

Papers [I] and [II] provide methods for formulating prior distributionsusing hierarchical meta-analyses of data from different stocks of the samespecies, or stocks of different but related species. Paper [I] focuses on im-proving the usage of information from data bases or literature, by learningfrom correlations that take place between biological parameters. Correlationsare considered to arise between biological parameters because of life historystrategies, as the energy needs to be allocated, for example, either to rapidgrowth or early maturation (Rochet 2000). Thus, the allocation of energycreates a statistical connection between parameters describing growth andmaturity. Correlations can also be learnt from related species, as those canbe expected to have similarities in their life history strategies. This is usefulespecially if the information available from the target species is sparse.

In paper [II], hierarchical meta-analysis is used for analyzing the stock-recruit relationship among a group of Atlantic salmon stocks. The paperaddresses the structural uncertainty arising from competing views about themathematical formula that best describes the connection between the spawn-ing stock size and the resulting number of offspring. The competing theoriesentail, for example, different views on how the high population densities af-fect juveniles chances to survive. Such theories, followed by different model

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choices can have a large impact on the management strategy of the stock.Furthermore, in paper [II] the stock specific maximum survival of eggs param-eters are estimated for stocks from which data is included in the modeling,and in addition also for another, or ’new’, Atlantic salmon stock, from whichno data is included. This predictive distribution for a new stock is estimatedbecause it can later be used as a prior distribution in a subsequent study,enabling learning from different studies.

Papers [III] and [IV] illustrate how Bayesian methods can be utilized whenanalyzing complex processes with vast amount or variety of data. The reviewpaper [III] illustrates the success story of Baltic salmon stock assessment, inwhich a Bayesian sequential model framework is utilized for combining allimportant sources of information together. As an example, it is shown howthe knowledge on salmon smolt abundance accumulates when informationfrom one analysis is transferred into another, adding new datasets into themodel at each step. The model framework demonstrated in [III] has beenformulated by others than myself (Mantyniemi and Romakkaniemi, 2002,Mantyniemi et al. 2004, Michielsens et al. 2008). However, I know thisparticular stock assessment framework very well because of the work I haveconducted within the past years in the International Council for the Explo-ration of the Seas (ICES) working group of Baltic salmon and sea trout (ICES2014a). Within the working group I have furthered the development of thestock assessment framework, for example, by including observation modelson river specific spawner count data (e.g. ICES 2011a) and by introducingannually varying maturation rates (ICES 2013).

The theme of modeling complex processes is continued in manuscript [IV],which takes the first steps in introducing probability based modeling into theanalysis of acoustic survey data from the Baltic International Acoustic Sur-vey (BIAS). A Bayesian hierarchical model is built to estimate the abundanceand the length/age composition of Bothnian Sea herring. The model is for-mulated from the perspective of the survey, accounting for the details of thesurvey design and data collection. The hierarchical structure is built bothover spatial and temporal dimensions, allowing information to flow betweengeographical areas and different years of survey.

2 Choices and sources of data

2.1 Data bases and literature

The FishBase (Pauly 1997, Froese and Pauly 2014) is a global species databasethat gathers data from published and grey literature about, for example,

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taxonomy, distribution, and life history parameters of all fish species. Thisdatabase is chosen as a data source in paper [I] because of the vast amountof biological data it contains, but also because it is easily available online foreverybody and thus it has a great potential for prior formulation in manystudies, regardless of available financial resources. To formulate an exam-ple on how correlations between biological parameters can be used to reduceuncertainty of the prior distribution, parameters of length-weight and length-fecundity relationships were extracted from the FishBase in paper [I] for threespecies from the Clupeidae family. These species are Atlantic herring (Clu-pea harengus), European pilchard (Sardina pilchardus) and Round sardinella(Sardinella aurita).

Length-weight and length-fecundity relationships are important compo-nents in stock assessments because many life history traits such as matura-tion, natural mortality and reproductive potential depend on the size of thefish. Thus, there is a trade-off in the allocation of energy of an individual thatcreates correlation between biological parameters (e.g. Rochet 2000). Otherbiological parameters, for example those related to the growth of the fish,could have been equally well used in this example, but the length-weight andlength-fecundity parameters were chosen as an example of a set of correlatedparameters in paper [I] because of the great variation in the amount of avail-able data between the three species. The length-weight relationship tendsto be more studied than the length-fecundity relationship, and thus learningfrom the correlations between these life history parameters has the potentialto improve our knowledge about length-fecundity. From the species perspec-tive, the availability of data also varies. Atlantic herring is more studied thanEuropean pilchard or Round sardinella and, therefore, it was considered asan opportunity to extend the knowledge from the data rich species to thedata poor.

The dataset in paper [I] contains information from 69 stocks, of which 29are of Atlantic herring, 22 of European pilchard and 18 of Round sardinella.Length-fecundity information is available, however, only for 8 stocks of At-lantic herring, 1 stock of European pilchard and 1 stock of Round sardinella.

The stock-recruitment data on 9 Atlantic salmon stock analyzed in study[II] is identical to those analyzed in Michielsens and McAllister (2004). Orig-inally these data was gathered by the authors of that paper from journalarticles and research documents in French and in English. The data consistsof annual spawning stock abundances and number of smolts (juveniles thatbegin their migration from river to sea) resulting from those spawners threeyears later.

Two of the nine datasets were in eggs-to-adults format instead of eggs-to-smolts format and were transformed by Michielsens and McAllister (2004)

6

to corresponding format assuming 10% survival rate from smolts to adults.In total, this dataset consists of 143 data points.

2.2 International Council for the Exploration of the

Seas (ICES)

In 1997, the Salmon Action Plan (IBSFC and HELCOM, 1999) set the inter-national long-term management goals for Baltic salmon stocks. As one of themost important goals was to ensure that each wild salmon population attainsat least 50% of its potential smolt production capacity by 2010, the simple,deterministic statistical tools used until then for stock assessment were nolonger sufficient. Varis and Kuikka (1997) first applied Bayesian inference toBaltic Salmon stock assessment by using a spreadsheet software and a modi-fication of a Bayesian net including population equations. After that, a wideframework of hierarchical Bayesian models has been developed to combineinformation from the data rich and data poor stocks together, in order to ob-tain the big picture of the status of the wild salmon in the Baltic. Today thisstock assessment methodology combines data from various sources into a lifehistory model that covers time series of almost 30 years and 15 wild Balticsalmon stocks at the Gulf of Bothnia and Baltic main basin (Michielsens etal. 2008, ICES 2014a).

The example in the review paper [III] illustrates how the knowledge aboutannual smolt abundance gets updated in the stock assessment when sequen-tial modeling steps are taken as follows: from the smolt mark-recaptureanalysis to a hierarchical model that combines the smolt estimates with thejuvenile density data sets, and further with all the other data sets that bringinformation about the strength of the smolt cohorts during later life stages.Smolt mark-recapture and electrofishing datasets are collected nationally inFinland and Sweden from the wild salmon rivers as a part of European Com-missions Data Collection Framework (DCF, EU 2008). The other datasetsused as input to the life history model contain, for example, river specificspawner counts, catch and effort information from the offshore and coastalfisheries, and Carlin tag recaptures (ICES 2014a).

The Baltic International Acoustic Survey (BIAS) is carried out annuallyas a part of the DCF to estimate the herring and sprat resources in the BalticSea. The survey gathers information about the total number and biomassof fish using echosounder and samples the species composition as well as thelength/age composition with trawl hauls (ICES 2011b). This information iscurrently analyzed using point estimation, providing additional informationabout abundance and length/age composition for the assessment of herring

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and sprat stocks (ICES 2014b).The analysis of paper [IV] covers data from BIAS in 2007-2012 from the

Bothnian Sea region (ICES sub-division 30). The acoustic data is averaged sothat 1 nautical mile of the cruise track gives one data sample. Contrary to thecurrent ICES methodology, only trawl data from night trawl hauls is takeninto account in paper [IV] for calculating the species and length compositions.This is because fish of different sizes are known to be distributed unevenlybetween various depth layers during day time, but to gather and mix nearthe surface at night (Cardinale et al. 2003). Consequently, usage of day timetrawl data would require separate modeling of different depth layers, whichwould be a very heavy computational task. Thus, the day time trawl datahas been omitted from the analysis. However, because the catch samplingfor age determination is length group based and is not intended to representthe age composition of the total population, catch samples from both nightand day trawl hauls are used in the estimation of the age composition ofBothnian Sea herring.

3 Methods

3.1 Bayesian inference

A fundamental feature of Bayesian inference is that probability is consideredto exist only as a subjective degree of belief (De Finetti 1975). Thus, theoutcome of statistical inference is considered subjective, depending on theperson implementing the analysis. This means that two experts may dis-agree on the results of their inference even if they observe the same data,because their background knowledge and/or views on how the data shouldbe interpreted, differs.

Some see the subjectivity as an argument against the Bayesian branchof statistics as a whole, claiming that science should be purely objective.In such arguments it is often forgotten that also the traditional statisticalinference requires subjective decisions, such as the choice of a likelihoodfunction. Data, once observed, can be considered objective, but when thatdata is interpreted and turned into knowledge about the subject studied, theobjectivity cannot be maintained. Instead of striving for objectivity, it ismore important to be able to justify the choices that have been made in anhonest and transparent way.

Bayesian inference is essentially a learning process where the initial knowl-edge (prior distribution) is updated with the most recent information (data,interpreted via the likelihood function) and these together form the new

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understanding about the phenomenon of interest (posterior distribution).Mathematically, this process is implemented first by setting up a joint prob-ability distribution for the observable data, y, and parameters of interest, θ.This can be obtained as the product of the prior distribution, which describesthe initial knowledge of the parameters (before the data were collected), andthe conditional distribution of the observations given the parameters, rep-resenting the interpretation of data. The prior distribution is a probabilitydistribution, and it is denoted by p(θ). The likelihood of the parameters giventhe data is proportional to the probability of the data given the parameters,p(y|θ) (Gelman et al. 2004). Thus, the joint probability distribution of thedata and the parameters is:

p(θ, y) = p(θ)p(y|θ).

Conditioning on the known y and using Bayes rule, we get the posteriordistribution:

p(θ|y) =p(θ)p(y|θ)

p(y),

where p(y) =∫p(θ)p(y|θ)dθ is the integral over all possible values of θ. Thus,

the posterior distribution expresses the new understanding about the phe-nomenon when the data, interpreted via the likelihood function, is combinedwith the initial knowledge.

The prior distribution is a component of statistical inference that onlyexists in the Bayesian framework. Its purpose is to be a synthesis of theavailable background information, i.e. the knowledge we had before collectingnew data. Because of the subjective nature of knowledge, the choice of a priordistribution may differ between experts. For instance, a statistician can havea very different idea about the possible values of a biological parameter of aspecific species of fish than a biologist whose career has been spent studyingthat species. On the other hand, views of different biological experts mayalso be combined into a single prior distribution, covering a wider range ofopinions that influence the range of plausible values (Uusitalo et al. 2005).Besides using expert knowledge, prior distributions can be formulated withany type of information coming from the literature (Vanpaemel 2011), variousdatabases (Froese et al. 2014), or previous Bayesian analyses (Michielsensand McAllister 2004).

The Bayesian inference has also been historically called as the inverse

probability of causes (e.g. McGrayne 2012), since it makes it possible toanswer questions about the probability of a phenomenon given the observeddata. In contrast, in the traditional (or frequentist) branch of statisticalinference, statements can only be made about the expected relative frequency

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of an outcome if the same experiment would be repeated infinitely manytimes. Because people tend to be interested in the probability that somethinghappens, or in finding out the probabilities of different potential reasonsbehind an observed event, results from the traditional statistical inferenceare also often misinterpreted as being probabilities (which is only the case inthe Bayesian framework).

3.2 Hierarchical models

Bayesian hierarchical models offer an efficient way to jointly analyze datafrom various sources that essentially tell us about the same phenomenon.Hierarchical models are also a practical tool for formulating prior distribu-tions (Gelman et al. 2004).

Suppose we have a set of fish stocks j = 1, . . . , n that are of same speciesand that we wish to learn about the biological parameters θ = (θ1, . . . , θn)of those stocks. These stocks may live in different geographical areas and indifferent environmental conditions, which implies that the values of θ are notidentical for the stocks, but since they are of the same species it is realisticto assume that those have something in common. If no feature is knownthat would distinguish any of the θj ’s from any of the others, and their jointdistribution is considered invariant to the permutation of the indexes, we canassume the parameters θ1, . . . , θn are exchangeable.

The exchangeability assumption is a subjective choice that reflects thelevel of knowledge of the person conducting the analysis. One analyst mightassume a set of parameters as exchangeable, whereas another analyst mighthave more information and instead consider those as conditionally exchange-

able. Conditionally exchangeable parameters are exchangeable in the residualvariation that remains after part of the variation has been explained with acovariate. For example, latitude can be such a covariate if the analyst be-lieves that the geographical location of the stock has a specific influence onthe value of the parameter of interest.

Because of the assumed exchangeability, each θj can be considered to bedrawn as an independent sample from the same prior distribution conditionedby a set of common hierarchical parameters φ:

p(θ|φ) =

n∏j=1

p(θj |φ).

As φ is usually not known, it will be given a prior distribution p(φ). Data islinked to θ through a likelihood function p(y|θ).

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In addition, hierarchical models enable us to make predictions of thedistribution of unknown parameters. If we wish to predict the parameter θ fora new stock that can be considered as exchangeable with the previous stocks,we can predict the distribution of θnew by integrating over the posteriordistribution of the hierarchical parameters φ:

p(θnew|y) =

∫p(θnew|φ)p(φ|y)dφ.

This predictive posterior distribution can be used as an informative priordistribution in a subsequent study.

Hierarchical modeling is utilized in all papers [I]-[IV] of this thesis. Inpaper [I], a hierarchical model is built between four biological parameters(describing length-weight and length-fecundity relationships) of three relatedspecies. The point estimates collected from the literature (FishBase; Froeseand Pauly 2014) for these biological parameters are treated as observed quan-tities. The data points are logarithm transformed, giving a range of valuesfrom potentially the entire real line, and those are assumed to follow a mul-tivariate (four-dimensional) normal distribution. The expected values of theparameters are further assumed to be normally distributed but unknown,and hierarchical priors are given to them, enabling learning between species.The variance-covariance matrix is given an inverse-Wishart prior distribu-tion, stating that the true variance-covariance structure is not known, but isdesired to be estimated. Correlations between the biological parameters ofthe length-weight and length-fecundity relationships are set to be commonfor all species. This makes it possible to learn about the correlations fromthose species from which plenty of data exists and transfer that knowledgeto species for which data is sparse.

A hierarchical structure is assumed both for the slopes at the origin of thestock-recruit function and for the model choice for a group of Atlantic salmonstocks in paper [II]. As the spawning stock sizes are considered as the num-ber of eggs, the slope at the origin parameter corresponds to the maximumsurvival of eggs, i.e. survival of eggs without density dependent mortality.The survival of eggs is assumed exchangeable among different salmon stocksfor computational simplicity. It is acknowledged, though, that the expertson Atlantic salmon might know of features that would, for example, dividethe stocks into geographical groups and, thus, be instead exchangeable con-ditionally on latitude. A predictive posterior distribution for the maximumsurvival of eggs parameter is estimated for a new Atlantic salmon stock; thisdistribution could be used as a prior in other studies (e.g. in Baltic salmonstock assessment [III]).

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The hierarchical structures in papers [III] and [IV] are described in sec-tions 3.4 and 3.5, respectively.

3.3 Structural uncertainty

The need to account for structural uncertainty arises when it is not clearhow a mathematical model should be formulated to best represent the phe-nomenon of interest. For example, experts may express competing viewsabout the nature of a biological process, each leading to very different modelformulations. On the other hand, even when there is not any doubt aboutthe biological process itself, different formulations of the same model canstill be considered. It may also be that the uncertainty about the possibleoutcomes is best represented by a mixture of several models, or by statingthat the true model resembles one or more functional forms and that morevariation is needed around those forms (Munch et al. 2005).

For stocks of Atlantic salmon, it is often discussed whether the Beverton-Holt or Ricker model is more suitable for describing their stock-recruit re-lationship (e.g. Hilborn and Walters 1992). Both models may fit the samedata well, but the management decisions can differ heavily depending onthe model choice. Eventually the choice of the model should depend onthe knowledge about the biology of a fish stock: With the Beverton-Holtmodel it is assumed that the number of recruits increases at first rapidlywith increasing spawning stock size and then slows down as the recruitmentapproaches the carrying capacity due to density dependent mortality. TheRicker model assumes similar behavior of the recruitment at low spawningstock sizes, but the recruitment then decreases after carrying capacity is met.This indicates that competition is so fierce that the recruitment suffers fromsignificant losses unless the spawning stock size is limited to being close tothe level that provides maximum recruitment. For example, differences inthe habitat quality or the amount of suitable spawning sites can influence thesurvival of eggs or juveniles and make the Ricker model a more biologicallyrealistic choice compared to Beverton-Holt (Aas et. al 2011).

The third commonly used model for the stock-recruit relationship is thehockey stick model (Barrowman and Myers, 2000). As the name suggests, ina hockey stick model the recruitment increases linearly until carrying capacityis met, after which it stays constant. Although the hockey stick model isnot biologically very plausible, it is often used because of its computationalsimplicity. Also, it is considered to give more conservative estimates for themaximum survival of eggs compared to Beverton-Holt and Ricker models.

In paper [II], the above three alternative models are considered as optionsfor each of the 9 Atlantic salmon stocks using the approach of Bayesian

12

model averaging (BMA, Hoeting et al. 1999). BMA is implemented with themethod of Carlin and Chib (1995), where an integer valued parameter is setto index the model collection. An uninformative hierarchical Dirichlet priordistribution, common to all the stocks in the analysis, is assumed for thestock specific model choices and for the parameters of the stock-recruitmentrelationships. As the parameters of the prior distribution get updated withthe data from the stocks that have lots of information about the shape of thestock-recruitment curve, the hierarchical structure transfers this informationto the stocks that do not contain much of such information. Even thoughequal prior weights are given for all three models, those could also be setunequal to emphasize different prior beliefs about how probable one modelis compared to another.

The paper [II] results in posterior probabilities for each of three stock-recruitment models. These probabilities could further be used as prior dis-tributions in a subsequent study if there is a need to account for structuraluncertainty.

3.4 Sequential models

Sequential models are useful when the model framework is computationallytoo complex to be run within a single model. Thus, two or more modelsare run separately, and information from one model is transferred to anotherusing, for example, likelihood approximation (e.g. Michielsens et al. 2008).Review paper [III] describes how uncertainty about the annual salmon smoltabundance estimates in Baltic salmon stock assessment decreases when dif-ferent types of information are included into the sequential model framework.Figure 1 illustrates these modeling steps and the related data sources.

First, data from a smolt mark-recapture study (Mantyniemi and Ro-makkaniemi 2002) is analyzed alone for each river from which such dataexists, estimating the annual strength of the smolt run in that particularriver. Secondly, a hierarchical model is built in a so-called river model (ICES2004) across a group of stocks, connecting the information from the smoltmark-recapture analysis with the electrofishing data that consists of juve-nile densities at several areas across the rivers. As the smolt mark-recapturedata is not available from all rivers or years, this hierarchical model makesit possible to learn from the connections between juvenile densities and thestrengths of the following smolt runs. Predictive posterior distributions areestimated for the smolt abundances for rivers/years from which only elec-trofishing data is available. As a third step in this sequential analysis, theposterior and predictive posterior distributions of smolt abundances obtainedfrom this analysis are transferred to the full life history-model (Michielsens et

13

Sea

Smolt mark-

recapture model

River model

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all rivers)

Catch data

Tag-recapture data

Etc.

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Etc.

Stock specific

smolt abundance

estimates

Rivers

Figure 1: Illustration of the Baltic salmon life cycle and sequential modelingsteps to estimate the stock specific smolt abundances. The dashed arrowsrepresent the input datasets associated with each model, whereas the dottedarrows represent the smolt abundance estimates that are transferred fromone model to another via likelihood approximation. (See paper [III] for moredetails.)

al. 2008), which deals with the entire life cycle of salmon, by using likelihoodapproximations.

In the full life history model, the smolt abundance estimates are againupdated by new sources of data, including data on Carlin tag mark-recapturestudies and annual stock specific strengths of the spawning runs. Becausemany parameters of the life cycle (for example, marine survival and rate ofmaturation at each sea age) are assumed common for all the stocks, the stockspecific smolt abundance estimates are influenced not only by the informationof the stock in question, but also by the information from other stocks. Forexample, large observed spawning run in one river may indicate high survivalfor one or several smolt cohorts, which may again update the smolt estimatesof other stocks.

14

3.5 Hierarchical spatial models

In addition to estimating population abundances, also the geographical lo-cation of the population at given time may be of interest. If spatial data onthe geographical distribution of the species is available, there is often a desireto use that data to find out how this distribution changes in time. Spatialmodels enable estimation of quantities within a grid of areas on a map, and ahierarchical structure between the areas makes it possible to expand knowl-edge from areas from which data are available into areas that are poorly,or not at all, studied. In addition, hierarchical spatial models can also bedefined continuously in space (e.g. Agarwal et al. 2005).

In paper [IV], a spatial hierarchical model is built for the Baltic Interna-tional Acoustic Survey over the Bothnian Sea region, in order to estimate theabundance, length and age composition of the Bothnian Sea herring stock. Inthis model, the Bothnian Sea is divided into 28 rectangles, and hierarchicalstructures are specified both across the rectangles and inside each rectan-gle, transferring information from the areas covered by the survey to areasoutside the survey track. The distribution of fish across the rectangles, aswell as inside each rectangle, is assumed to be Dirichlet. An overdispersionparameter is set to cover the extra variation that can arise from the school-ing behavior of the fish, compared to the variation assuming independenceof fish locations. Similarly, the distribution of the fish within a rectangle,considering the areas covered by pieces of cruise track and areas outside thesurvey track, is assumed to be Dirichlet.

Observation models for data from the trawl catches are specified forspecies composition, for length composition of herring and other species,and for age composition of herring. The species composition is accountedfor by assuming that the observed proportion of herring in each statisticalrectangle is Beta- distributed and linking the expected value of that distri-bution with the annual proportion of herring in the total Bothnian Sea. Theobserved length compositions of herring and other species are assumed tofollow the Dirichlet-Multinomial distributions. In a similar manner, the agecomposition of herring is assumed to be Dirichlet-Multinomial distributed.The length class specific sampling scheme for ages in the catch sampling isaccounted for when estimating the age composition of the herring population.

3.6 Software

The Bayesian models in this thesis have been analyzed using Markov ChainMonte Carlo (MCMC) simulation methods with WinBUGS/OpenBUGS (Lunnet al. 2009; papers I and III), or with JAGS (Plummer 2003; papers II and

15

IV) software. In study [IV] Hamiltonian Monte Carlo methods were alsotested using Stan (Stan Development Team 2014) to increase computationalefficiency, but unfortunately convergence of parameters was not achieved.Convergence of the MCMC chains in all studies has been monitored usingthe Gelman-Rubin shrink factor diagnostics and/or visual inspection of thechains.

4 Results

4.1 Learning from correlations between biological pa-rameters

Estimated posterior distributions in paper [I] indicate that the parametersof length-weight and length-fecundity relationships are strongly correlatedamong the three studied species of the clupeidae family (Fig 2). The 95%posterior probability intervals (PI) for all six pairwise correlations are eitherabove 0.5 or below -0.5.

The species specific mean estimates from the correlation model are com-pared to the estimates obtained from the same hierarchical model but withoutthe correlation structure (Fig 3). It can be seen that utilizing correlation re-duces uncertainty in the length-fecundity estimates, especially if data fromboth biological relationships is available from the same stocks. Thus, theuncertainty is strongly reduced for the length-fecundity parameters of Euro-pean pilchard but not so much for Round sardinella, as for the latter speciesdata of length-fecundity and length-weight relationships is not available fromthe same stocks.

Posterior estimates of the length-weight relationship parameters are rathersimilar for all species between the correlation model and the hierarchicalmodel without correlation (Fig 3), suggesting that utilization of correlationsdoes not make a much difference if there is a large amount of data availablefor all parameters.

4.2 Accounting for model uncertainty in the stock-recruit relationship of Atlantic salmon

The probabilities of three alternative models for the stock-recruitment re-lationship (Beverton-Holt, Ricker and hockey stick) are calculated for nineAtlantic salmon stocks using the method of Bayesian model averaging (pa-per [II]), and one clear winner is found. The posterior distribution for theproportion of stocks with Beverton-Holt as the true model is estimated to

16

−1.0 −0.9 −0.8 −0.7 −0.6 −0.5

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cor(bw,bf)−1.0 −0.9 −0.8 −0.7 −0.6 −0.5

05

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30

cor(af,bf)

Figure 2: Prior (dashed line) and posterior (solid line) distributions for thepairwise correlations between parameters of the length-weight relationship(aw, bw) and length-fecundity relationship parameters (af , bf ) of three speciesfrom the clupeidae family (Atlantic herring, European pilchard and Roundsardinella).

17

−5.6 −5.2 −4.8 −4.4

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Figure 3: Prior and posterior distributions of mean parameters of length-weight relationship (µaw , µbw) and length-fecundity relationship (µaf , µbf ) forAtlantic herring, European pilchard and Round sardinella species. Dottedline illustrates the prior distribution, whereas thin solid line illustrates theposterior distribution based on the model without correlation and bold solidline the posterior distribution based on the correlation model, respectively.

18

have median of 0.75 and 95% PI of [0.34,0.96]. For individual stocks, three ofthe nine stocks have more than 0.95 probability for Beverton-Holt whereasfor the rest of the stocks this probability varies between 0.74 and 0.87. Thepredictive probability for an unknown salmon stock to have a Beverton-Holtmodel is estimated to be 0.72, whereas the corresponding probabilities are0.11 for the Ricker model and 0.17 for the hockey stick model. Figure 4illustrates the estimated stock specific stock-recruitment relationships as amixture of the alternative models, where each curve represents one randomdraw from the joint posterior distribution of the model parameters, includingthe model choice.

The predictive distribution for the maximum survival of eggs parameterof a new salmon stock (from which no data is included in this analysis)is estimated to have median of 0.05 and 95% PI of [0.01,0.51]. For thenine Atlantic salmon stocks whose data is analysed, posterior medians of themaximum survival vary between 0.02 and 0.10 (Fig 5), with River Ellidaarhaving the highest range of uncertainty in the estimates (95% PI [0.01,0.53])and Little Codroy River having the smallest range of uncertainty (95% PI[0.04,0.19]), respectively.

4.3 Updating knowledge about Baltic salmon recruit-

ment with a sequential analysis

The annual smolt abundances in river Tornionjoki (paper III) are estimatedat three consecutive steps when the sequential model framework is applied inthe Baltic salmon stock assessment (Fig 6). In these models, biological back-ground knowledge is combined with the data to provide probability estimatesfor the smolt abundances.

The smolt mark-recapture model (Mantyniemi and Romakkaniemi 2002)provides the first estimate about the smolt abundance on river Tornionjokibased on the smolt trapping data. This estimate has the greatest uncer-tainty among the three models. Next, the estimates from the smolt mark-recapture model are updated in the river model (ICES 2004), which usesjuvenile density information across 15 rivers at Baltic Sea. Finally, the fulllife history model (Michielsens et al. 2008) updates the river model estimateson the smolt abundance utilizing, for example, the spawner count, fisheriesand Carlin-tag-recapture datasets as well as the model assumptions aboutthe biology of salmon. The smolt abundance estimates do not differ greatlybetween the river model and the full life history model, because most ofthe information about the smolt abundance is already covered by the firsttwo models. Information about the adult abundances may have an effect

19

Figure 4: Estimated stock-recruit relationship for nine Atlantic salmon stockas a mixture of three alternative models (Beverton-Holt, Ricker and hockeystick). Each curve represents a random draw from the joint posterior distri-bution of model parameters.

20

0.00 0.05 0.10 0.15 0.20 0.25

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Figure 5: Prior and posterior estimates for maximum survival of eggs fornine Atlantic salmon stocks and a predictive distribution for another salmonstock, from which no data is included in the analysis.

21

2000 2005 2010

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00

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olts

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thou

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Figure 6: The estimated smolt abundance in river Tornionjoki after sequen-tial modeling steps of smolt mark-recapture model, river model and the fulllife history model.

backwards to the previous smolt abundance, but often such data will ratherupdate the estimated mortality rates. For example, high observed spawningrun size can either be interpreted so that the smolt cohorts contributing tothe spawning run must have been larger compared to what smolt and juve-nile studies suggest, or such data can be seen as an evidence about highersmolts-to-adults survival. The latter interpretation usually more plausible,since prior knowledge about the survival rates is rather uninformative.

4.4 Introducing uncertainty into the estimates of BalticInternational Acoustic Survey

Because of the computational problems and poor convergence in some of theestimates, the posterior results of the acoustic survey model [IV] illustratedare preliminary and should be treated with caution. The convergence isproblematic especially for the estimated abundance and into lesser extent tothe estimated length-age composition, but the possibility that changes in theestimated abundance would also affect the length-age composition cannot beruled out.

The estimated length compositions of herring in 2009-2012 suggest ahigher share of larger herring (length groups 17.5-20 cm and 20-22.5 cm)

22

and a lower share of smaller herring (length group 12.5-15 cm) comparedto the standard point estimates of the ICES working group WGBFAS (Fig7). The difference in length composition impacts the estimated abundances,which are systematically lower according to Bayesian estimates compared tothe standard estimates in all years 2009-2012 (Fig 8). However, the datasetsbehind the estimation procedures are not fully comparable, as the Bayesianestimates only contain data from the night trawl hauls.

Similarly to the estimates of annual length composition, estimated annualage composition also has a higher share of older herring (8+ year olds) andlower share of younger herring (1-2 year olds) compared to the standardestimates (Fig 9). The results of the two estimation procedures differ themost for 1 year olds in 2012: the Bayesian estimate for this age group in2012 is clearly higher than in earlier years, but not as high as the standardestimate suggest. It is interesting to note that in 2012 the financial resourcesof the survey were reduced to a half and, thus, the length of the cruise trackand the number of trawl hauls needed to be reduced by 50% compared toprevious years.

5 Discussion

This thesis provides some formal statistical tools for accounting for sourcesof uncertainty that are often ignored in fisheries stock assessments. Themethodological core of this thesis is based on Bayesian hierarchical modeling,as this allows both the synthesis and accumulation of knowledge. Modelstructures in papers [I], [II] and [IV] have not been published before, whereasthe model framework in [III] has been published in parts within the pasttwelve years (Mantyniemi and Romakkaniemi 2002; ICES 2004; Michielsenset al. 2008). There remain many ways to improve the approaches illustratedin this thesis, if additional resources for funding the work are available. Itshould be noted that these methods could be applied also to species otherthan fish. Especially the approaches of papers [I] and [II] are very general andcould also be useful in stock assessments of bird, mammal or insect species.

The correlation model in paper [I] shows how correlations among bio-logical parameters can benefit the prior formulation of stock assessments.Despite the fact that model structure in [I] is simple and intuitive, it hasnot been used before to improve the estimates of biological parameters. Ac-counting for correlations can be advantageous, especially if a stock assessmentneeds to be provided for a stock that is poorly known, as usually in such casethe resources to collect new data are limited. Because the correlations arisefrom different life history strategies, additional information is likely to be

23

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Figure 7: Estimated annual length composition of Bothnian Sea herringin 2009-2012. Dots and lines illustrate the medians and 95% probabilityintervals of Bayesian estimates, whereas the standard point estimates areillustrated with triangles.

24

10 15 20 25 30

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Figure 8: Estimated posterior density of the annual abundance of BothnianSea herring in 2009-2012. The standard point estimates are illustrated withvertical lines. Note that x-axis in 2011 graph differs from the others.

25

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Figure 9: Estimated annual age composition of Bothnian Sea herring in 2009-2012. Dots and lines illustrate the medians and 95% probability intervals ofBayesian estimates, whereas the standard point estimates are illustrated withtriangles.

26

found from any set of species. Whereas the dataset in [I] consists of onlythree species, it is better to have a larger number of species in an analysis,if possible. This is preferable especially if there is a need to provide a pre-dictive distribution for an unknown species. If data from a wide selection ofspecies is included in the analysis, the predictive distribution is less likely tobe driven by the life strategy of a single species and more likely to reflect awider range of biological knowledge.

Difficulty in finding data on several life history parameters for the samestock can hinder the usage of the model [I] in practice. Data itself maybe widely available, but it can be unclear which datasets can be consideredto represent the same stock. This is troublesome because data is neededfrom the same stocks in order to estimate correlations. In FishBase (Froeseand Pauly, 2014) each dataset is connected to a stock description based onthe original publication, but problems arise from not knowing whether twodescriptions may or may not indicate the same stock. The definition ofa stock itself can also be vague, and it may be difficult to distinguish onesub-population from another. Recently, the FishBase staff has included stockidentification numbers for some species, but understandably doing this for alldatasets that exist in the database is an enormous task. Thus, developmentsin data bases may increase the usage of model [I] in the future.

Our approach in paper [II] is the first to use hierarchical modeling toaccount for structural uncertainty in a relationship between spawning stocksize and resulting offspring recruitment. The biological realism of a stock-recruit analysis is increased by acknowledging that it is not clear which oneof the functional forms (Beverton-Holt, Ricker or hockey stick) would be thebest in describing the future behavior of the Atlantic salmon stocks. Similarwork was also conducted by Hillary et al. (2012), who evaluated probabilitiesof Ricker and Beverton-Holt types of models for a set of herring stock-recruitdata using a Shepherd model that takes values above or below 1 (for a certainparameter) depending on which type of model best fits the data. Comparedto their approach, the method in [II] is more general since the models donot need to be of the same functional form. This enables estimation ofprobabilities for practically any type of model structure.

The approach of accounting for model uncertainty in paper [II] could bedeveloped further by incorporating explanatory variables or informative priordistributions for the model choice. Also, the approach could be expandedwith the method of Munch et al. (2005), allowing each alternative model toincorporate a Gaussian process prior. This would suggest that the detailsabout the shape of each function would not be precisely known, althoughthey would resemble known forms. On the other hand, the simplifying as-sumption that the functional forms of the stock-recruit relationship can be

27

considered as exchangeable between different stocks, may require more care-ful consideration. This assumption implies that nothing is known that wouldmake one stock more likely to have e.g. a Ricker as the correct functionalform than some other stock. However, differences between stocks could arise,for example, from the habitat conditions in different rivers and, if such infor-mation existed, the assumption about exchangeability should be revised. Itmust be acknowledged, though, that exchangeability reflects a state of mindand not a state of nature. Thus, the model structure chosen will depend onthe level of knowledge that the person implementing the model holds.

We also claim in paper [II] that the maximum survival of eggs parametershould be used as a key parameter of a stock-recruit relationship instead ofsteepness, if the purpose of the stock-recruit meta-analysis is to provide apredictive prior distribution for a subsequent study. This change would im-prove the biological realism of the stock-recruit analyses in two ways: First,by avoiding estimation of the steepness parameter, the need to make as-sumptions about life history parameters other than those taking place be-tween spawning and recruitment, is removed. Second, as the estimation ofmaximum survival of eggs parameter requires that the spawning stock sizeis considered in terms of eggs, there is an opportunity to better representthe reproductive potential of the spawning stock. This is because the cur-rently widely used spawning stock biomass estimate (SSB) does not accountfor temporal variability in fecundity due to, for example, parental size andresource variability (Marshall et al. 2003, Lambert 2008).

Paper [III] illustrates the success story of the Bayesian stock assessmentframework applied to Baltic salmon. The sequential estimation process ofsmolt abundance is used as an example of the benefits of this model struc-ture. There are, however, many other interesting details in this model frame-work, such as the estimation of annual post-smolt mortality rates or annualmaturation rates for fish of a certain sea age. The ability to combine infor-mation from different sources (e.g. tagging, catch and spawner count data)and to encapsulate the biological background information from literatureand experts (e.g. migration patterns, potential smolt production capacities)into the model structure is one of the greatest advantages of the Bayesianmodeling framework.

The model framework of the Baltic salmon stock assessment is still evolv-ing. One of the aims of the future model development is to re-parametrize theriver specific stock-recruit relationships by using maximum survival of eggs askey parameter of the Beverton-Holt function. Currently the parametrizationis based on steepness (Michelsens and McAllister, 2004), but the downsideof this approach is that the assumptions made about adult survival for At-lantic salmon stocks in the meta-analysis do not necessarily apply for Baltic

28

salmon stocks. Instead, results in paper [II] would provide the prior dis-tribution for the maximum survival of eggs parameter of the Beverton-Holtfunction, removing the need to assume anything about the adult survival inthe stock-recruitment relationship.

In paper [IV], first steps are taken towards introducing probability basedstatistical inference to the analysis of the Baltic International Acoustic Sur-vey data. The standard estimation method used by the ICES working groupWGBFAS is based on point estimation and is not able to reflect uncertain-ties that arise, for example, from the variation in the spatial coverage of thesurvey. The results based on the Bayesian model in [IV] suggest that theBothnian Sea herring stock would consist of somewhat larger herring, butthat the total herring abundance would be far lower compared to the ICEScurrent interpretation of the BIAS survey data.

The comparison between the standard ICES estimates and the new Bayes-ian estimates is hampered by the fact that the analyzed datasets are notidentical: the Bayesian model contains trawl data only from the night trawlhauls, whereas the standard estimates are based on both night and day trawldata. The reason for omitting the day trawl data from the Bayesian es-timation is that during daytime fish tend to be located in different waterdepths depending on their size (Cardinale et al. 2003), whereas at nightfish of different size gather together near the surface. Thus, the night timetrawl catches give a more representative sample of the size composition ofthe population if the water depth is not accounted for in the analysis. As thisissue is not accounted for in the standard estimation procedure, our Bayesianapproach may be more robust as it acknowledges the dial variation in thevertical distribution of herring and only uses night trawl data.

However, computational problems have so far prevented the results ofpaper [IV] from being used in the stock assessment of Bothnian Sea herring.The simulation run with two parallel chains in JAGS illustrated in [IV] takesone month to run, but MCMC chains of important parameters do not yetmix well and thus their convergence has not been met. Thus, the perfor-mance of the model and existence of potential faults needs to be studiedcarefully before the estimates can be utilized in the Bothnian Sea herringstock assessment.

Computational difficulties are undeniably the greatest hindrance in achiev-ing wider usage of Bayesian hierarchical models. Simple models such as [I]or [II] run quite fast, but the more complex or highly dimensional the modelsare the more time the simulation runs take. For example, the computationaltime of the stock assessment model of Baltic salmon [III] has reached itslimits with more than one month of run time, which is roughly the sametime which there is left to provide the assessment after annual updates on

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stock assessment datasets become available. This makes the assessment quitevulnerable to mistakes as there does not remain much time to deal with com-putational difficulties or to redo the analysis if errors in the input data arediscovered.

Computational problems often arise because the joint posterior distribu-tions are strongly correlated. The Gibbs sampling based software such asJAGS and BUGS move very slowly in a highly correlated parameter spaceand thus it takes a very long time to generate a representative sample fromthe target distribution. More efficient algorithms, for example the ones utiliz-ing Hamiltonian Monte Carlo sampling (Stan Development Team 2014), haveshown great promise in sampling from highly correlated parameter spaces,and are likely to replace older simulation methods in the future. Alterna-tively, custom written samplers could be designed to tackle sampling prob-lems, but this option is also costly as it requires both the expertise and thetime for dealing with each specific modeling case.

In addition to lengthy run times, another disadvantage of the complexBayesian models is that it can be quite difficult to explain the behavior andperformance of a model to non-statisticians, especially to those with verylittle previous experience in Bayesian modeling. As the number of details ina model increases, the longer it takes for someone to understand the model asa whole, and the fewer the people that have the time to learn all the nuances.This is one of the reasons why simple models are sometimes argued to bepreferable to complex ones (e.g. Hodges 2010, Wenger and Olden, 2012).However, if simple models are systematically preferred over complex ones,this may easily happen to the detriment of biological realism. The Balticsalmon case [III] is a great example on how the complexity of a model, ifarising from the biology, can provide important knowledge on the conditionof the stocks, contributing to their recovery.

Challenges in the fisheries stock assessment are not about to diminishin the future. Over-fishing of stocks, not only for human consumption butalso for fish meal production for farmed fish, threatens fish stocks globally(Naylor et al. 2000). The growing industry of aquaculture threatens alsostocks of migratory fish by increasing the amount of parasites and diseasesat migration routes (Brauner et al. 2012). Furthermore, the climate changeinduces constantly new threats to the fish stocks, and the fisheries managersshould be able to quickly react to the changes in order to secure the futureof the stocks (Ficke et al. 2007). To meet these challenges, the practicesof fisheries stock assessment will need to evolve accordingly. Bayesian hi-erarchical models offer a solid methodological background for this, enablingsynthesis and accumulation of a wide range of knowledge and even competingviews on reality. As the subjective nature of statistical inference is acknowl-

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edged, it becomes easier to honestly examine and account for the sources ofuncertainty that affect the results of a stock assessment.

Acknowledgements

First and foremost I want to thank my supervisors, Professor Samu Mantynie-mi and Professor Jukka Corander for their guidance, support and encourage-ment throughout this work. My warmest thanks go to Carmen Fernandezand Len Thomas for pre-examining my thesis, Professor Sakari Kuikka andProfessor Jaakko Erkinaro for providing support and advice as my follow-up team and for Professor Mikko Sillanpaa for agreeing to be my opponent.I owe my gratitude to Professor Sakari Kuikka for inviting me to join theFisheries and Environmental Management group (FEM) and for suggestingI could conduct my thesis in the ECOKNOWS project. These opportunitieshave enabled me to learn a lot and to network with many talented and en-thusiastic scientists. Furthermore, I want to thank Riitta Rahkonen and AriLeskela for arranging the funds which have enabled me to conduct this workat Game and Fisheries Research. Warmest thanks go to all my co-authors,Polina Levontin, Jarno Vanhatalo, Anna Gardmark, Olavi Kaljuste, TommiPerala, Juha Lilja and Andreas Bryhn for their fruitful collaboration, andto Polina Levontin and Paivi Laukkanen-Nevala for their valuable commentson the summary of this thesis.

I am indebted to Atso Romakkaniemi and Tapani Pakarinen who havebeen my mentors into the world of fisheries science and especially to salmon.Atso is the person to thank for initially drawing me into the Bayesian mod-eling, first by offering me a masters thesis project and a few years laterproviding an opportunity to join the ICES working group of Baltic salmonand sea trout, WGBAST. This was an extraordinary opportunity for a youngstatistician, which set my future as a Bayesian modeler. For ending up intothis path, I owe my gratitude also to Catherine Michielsens, the mother ofthe Baltic salmon stock assessment framework, who bequeathed this wonder-ful tool to me and taught me how to use it. Within the years, the work withBaltic salmon has taken me to many places and I want to thank all the pastand present colleagues in the WGBAST, including Johan Dannewitz, StefanPalm, Stig Pedersen, Piotr Depowski, Martin Kesler and Soile Kulmala. Maythere be many more stock assessments yet to come! Scenarios!

I have been fortunate to work with many people from different instituteswithin this work. I am extremely grateful for all the colleagues I have hada chance to meet and to learn from in the ECOKNOWS project, includ-ing Rebecca Whitlock, Anna Gardmark, Olavi Kaljuste, Andreas Bryhn,

31

Etienne Rivot, Felix Massiot-Granier, Jonathan White and Margarita Rin-con, to mention but a few. I wish to thank Inari Helle, Annukka Lehikoinen,Paivi Haapasaari, Kirsi Hoviniemi, Tommi Perala, Jarno Vanhatalo and MikaRahikainen from FEM group for all the peer support and inspiring discus-sions throughout the years. Huge, warm thanks belong naturally to all my(past and present) colleagues in Oulu Game and Fisheries Research, includingMaare, Riina, Ville, Panu, Jaakko, Aki, Tapio, Pekka, Samuli, Ilpo, Mikko,Alpo, Teukka, Mari, Salla, Jenni, Paivi and Antti. Because of you, there isnever such a thing as a boring day at the office! Furthermore, I am gratefulfor Professor John Mumford, Adrian Leach and Polina Levontin for arrangingme the opportunity to visit Imperial College London within my studies, andfor your company and many memorable moments during the visit. Moreover,I owe my deepest gratitude to my friend and colleague Polina for encourag-ing me to visit London time after time and especially for introducing me toworld of salsa, which has been one of the greatest sources of energy and joyin my life ever since.

I am extremely grateful to all my friends who have always taken carethat laughter and good food do not run out when we meet. I thank all mysalsa friends and the wonderful Latin ladies at the Cross Move Companywho have ensured that the counterbalance for the scientific work is alwaysnear when needed. Special thanks go to my dear friends Taarna and Mikaelfor providing me help and guidance in the academic jungle from the verybeginning, and to Tiina for being my trusted companion in philosophicaldiscussions, whimsical ideas and most of all in supportive friendship since1999.

Finally, I am thankful to my parents Kaija-Leena and Jouko, my brotherMauri, my grandma Sylvi and my dear spouse Petteri for your love, supportand understanding throughout this process. All my achievements would bemeaningless if I did not have you to share those with.

This work has mainly been funded by the Finnish Game and FisheriesResearch Institute, Finnish Doctoral Programme in Stochastics and Statistics(FDPSS) and Department of Environmental Sciences of the University ofHelsinki. A great part of the work has been carried out with funding fromECOKNOWS project (EU 7th framework programme). Financial help forfinishing this thesis was received also from the Finnish Concordia Fund.

References

Aas, O., Einum, S., Klemetsen, A., and Skurdal, J. 2011. Atlantic salmonecology. Wiley-Blackwell, Oxford.

32

Agarwal, D.K., Silander, J.A., Gelfand, A.E., Dewar, R.E., and Mickelson,J.G. 2005. Tropical deforestation in Madagascar: analysis using hierarchi-cal spatially explicit, Bayesian regression models. Ecol. Model. 185: 105-131.

Barrowman, N.J., and Myers, R. A. 2000. Still more spawner-recruitmentcurves: the hockey stick and its generalizations. Can. J. Fish. Aquat. Sci.57(4): 665-676.

Brauner, C. J., Sackville, M., Gallagher, Z., Tang, S., Nendick, L. and Farrell,A. P. 2012. Physiological consequences of the salmon louse (Lepeophtheirussalmonis) on juvenile pink salmon (Oncorhynchus gorbuscha): implicationsfor wild salmon ecology and management, and for salmon aquaculture.Philos. T. R. Soc. B. 367: 1770-1779.

Brodziak, J., and Legault, C.,M. 2005. Model averaging to estimate rebuildingtargets for overshed stocks. Can. J. Fish. Aquat. Sci. 62(3): 544-562.

EU 2008. Commission Regulation (EC) No 665/2008 of 14 July 2008. OfficialJournal of the European Union, 15 July 2008, L 186: 3-5.

Ficke, A. D., Myrick, C. A., and Hansen, L.J. 2007. Potential impacts ofglobal climate change on freshwater fisheries. Rev. Fish. Biol. Fisher. 17(4):581-613.

Froese, R., and Pauly, D. (eds). 2014. FishBase, World Wide Web electronicpublication www.fishbase.org (6/2014)

Froese, R., Thorson, J.T., and Reyes, R.B. 2014. A Bayesian approach forestimating length-weight relationships in shes. J. Appl. Ichtyol. 30: 78-85.

Carlin, B. P., and Chib, S. 1995. Bayesian Model Choice via Markov chainMonte Carlo methods. J. R. Stat. Soc. B. Met. 57(3): 473-484.

De Finetti, B. 1975. Theory of Probability, Vol 1-2. John Wiley & Sons Ltd.,Chichester.

Gelman, A., Carling, J.B., Stern, H.S., and Rubin, D.B. 2004. Bayesian DataAnalysis (2nd ed.). Chapman & HallCRC, Boca Raton.

Gislason, H., Daan, N., Rice, J.C., and Pope, J.G. 2010. Size, growth, temper-ature and the natural mortality of marine fish. Fish. Fish. 11(2): 149-158.

Haapasaari, P., Kulmala, S., and Kuikka, S. 2012. Growing into Interdis-ciplinarity: How to Converge Biology, Economics, and Social Science inFisheries Research? Ecol. Soc. 17(1): 6.

33

Harwood, J., and Stokes, K. 2003. Coping with uncertainty in ecologicaladvice: lessons from fisheries. Trends. Ecol. Evol. 18: 617-622.

He, X., Mangel, M., and MacCall, A. 2006. A prior for steepness in stock-recruitment relationships, based on an evolutionary persistence principle.Fish. Bull. 104(3): 428-433.

Hilborn, R., andWalters, C.J. 1992. Quantitative Fisheries Stock Assessment:Choice, Dynamics & Uncertainty. Chapman & Hall, New York.

Hill, S.L., Watters, G.M., Punt, A.E., McAllister, M.K., Le Quere, C., andTurner, J. 2007. Model uncertainty in the ecosystem approach to sheries.Fish. Fish. 8: 315-336.

Hillary, R.M., Levontin, P., Kuikka, S., Mantyniemi, S., Mosqueira, I., andKell, L. 2012. Multi-level stockrecruit analysis: beyond steepness and intomodel uncertainty. Ecol. Model. 242: 69-80.

Hoeting, J.A., Madigan, D., Raftery, A.E., and Volinsky, C.T. 1999. Bayesianmodel averaging: a tutorial. Stat. Sci. 14(4): 382-417.

Hodges, J.S. 2010. Are exercises like this a good use of anybody’s time?Ecology. 91(12): 3496-3500.

Hoshino, E., Milner-Gulland, E.J., Hillary, R.M. 2014. Why model assump-tions matter for natural resource management: interactions between modelstructure and life histories in fishery models. J. Appl. Ecol. 51(3): 632-641.

IBSFC and HELCOM. 1999. Baltic salmon rivers - status in the late 1990sas reported by the countries in the Baltic region. Technical report, TheSwedish Environmental Protection Agency, the Swedish National Board ofFisheries.

ICES 2004. Report of the Baltic Salmon and Trout Assessment WorkingGroup. ICES CM 2004/ACFM:23. Annex 2: Mantyniemi et al. Predictingsalmon smolt abundance using a simple hierarchical Bayesian model.

ICES 2011a. Report of the Baltic Salmon and Trout Assessment WorkingGroup (WGBAST), 22-30 March 2011, Riga, Latvia. Technical ReportICES 2011/ACOM:08, International Council for the Exploration of theSeas.

ICES 2011b. Manual for the Baltic International Acoustic Survey (BIAS).ICES WGBIFS report 2011, Addendum 2. International Council for theExploration of the Seas.

34

ICES 2013. Report of the Baltic Salmon and Trout Assessment WorkingGroup (WGBAST), 3-12 April 2013, Tallinn, Estonia. Technical ReportICES 2013/ACOM:08, International Council for the Exploration of theSeas.

ICES 2014a. Report of the Baltic Salmon and Trout Assessment WorkingGroup (WGBAST), 26 March - 2 April 2014, Aarhus, Denmark. TechnicalReport ICES 2014/ACOM:08, International Council for the Exploration ofthe Seas.

ICES 2014b. Report of the Baltic Fisheries Assessment Working Group (WG-BFAS). 3-10 April 2014 ICES HQ, Copenhagen, Denmark. ICES CM2014/ACOM:10.

Jiao, Y., Cortes, E., Andrews, K., and Guo, F. 2011 Poor-data and data-poor species stock assessment using a Bayesian hierarchical approach. Ecol.Appl. 21(7): 2691-2708.

Kuparinen, A., Mantyniemi, S., Hutchings, J.A., and Kuikka, S. 2012. In-creasing biological realism of sheries stock assessment: towards hierarchicalBayesian methods. Envir. Rev. 20(2): 135-151.

Lambert, Y. 2008. Why should we closely monitor fecundity in marine fishpopulations? J. Northw. Atl. Fish. Sci. 41: 93-106.

Lu, G., and Ades, A. 2009. Modeling between-trial variance structure inmixed treatment comparisons. Biostatistics. 10:4, 792-805.

Lunn, D., Spiegelhalter, D., Thomas, A., and Best, N. 2009. The BUGSproject: Evolution, critique, and future directions. Stat. Med. 28: 3049-3067.

Marshall, C.T., O’Brien, L., Tomkiewicz, J., Koster, F.W., Kraus, G.,Marteinsdottir, G., Morgan, M.J., Saborido-Rey, F., Blanchard, J.L.,Secor, D.H., Wright, P.J., Mukhina, N.V., and Bjornsson, H. 2003. De-veloping alternative indices of reproductive potential for use in sheriesmanagement: case studies for stocks spanning an information gradient.J. Northw. Atl. Fish. Sci. 33: 161-190.

McAllister, M.K., and Kirchner, C.H. 2001. Development of Bayesian stockassessment methods for Namibian orange roughy Hoplostethus atlanticus.S. Afr. J. Marine. Sci. 23: 241-264.

McGrayne, S.B.. 2011. The theory that would not die. Yale University Press.New Haven & London.

35

Myers, R.A., and Mertz, G. 1998. Reducing uncertainty in the biological basisof sheries management by meta-analysis of data from many populations: asynthesis. Fish. Res. 37: 51-60.

Michielsens, C.G.J., and McAllister, M.K. 2004. A Bayesian hierarchical anal-ysis of stockrecruit data: quantifying structural and parameter uncertain-ties. Can. J. Fish. Aquat. Sci. 61(6): 1032-1047.

Michielsens, C.G.J., McAllister, M.K., Kuikka, S., Pakarinen, T., Karlsson,L., Romakkaniemi, A., Pera, I., and Mantyniemi, S. 2006. A Bayesianstatespace mark-recapture model to estimate exploitation rates in mixedstock sheries. Can. J. Fish. Aquat. Sci. 63: 321-334.

Michielsens, C.G.J., McAllister, M.K., Kuikka, S., Mantyniemi, S., Ro-makkaniemi, A., Pakarinen, T., Karlsson, L., and Uusitalo, L. 2008. Com-bining multiple Bayesian data analyses in a sequential framework for quan-titative sheries stock assessment. Can. J. Fish. Aquat. Sci. 65: 962-974.

Mullon, C., Freon, P., and Cury, P. 2005. The dynamics of collapse in worldfisheries. Fish. Fish. 6: 111-120.

Munch, S.B., Kottas, A., and Mangel, M. 2005. Bayesian nonparametricanalysis of stock-recruitment relationships. Can. J. Fish. Aquat. Sci. 62(8):1808-1821.

Mantyniemi, S., and Romakkaniemi, A. 2002. Bayesian mark-recapture esti-mation with an application to a salmonid smolt population. Can. J. Fish.Aquat. Sci. 59: 1748-1758.

Mantyniemi, S., Romakkaniemi, A., Uusitalo, L. and Arjas, E. 2004. Predict-ing salmon smolt abundance using a simple hierarchical Bayesian model.ICES WGBAST report 2004, Annex 2. International Council for the Ex-ploration of the Seas.

Naylor, R.L., Goldburg, R.J., Primavera, J.H., Kautsky, N., Beveridge,M.C.M., Clay, J., Folke, C., Lubchenko, J., Mooney, H., and Troell, M.2000. Effect of aquaculture on world fish supplies. Nature. 405: 1017-1024.

Parma, A.M. 2002 Bayesian approaches to the analysis of uncertainty in thestock assessment of pacific halibut. Am. Fish. S. S. 27: 113-136.

Pauly, D. 1997. The Science in FishBase. EC Fisheries Cooperation Bulletin,10(2): 46.

Pinheiro, J. C., and Bates, D.M. 1996. Unconstrained parametrizations forvariance-covariance matrices. Statist. Comput. 6: 289-296.

36

Plummer, M. 2003. JAGS: a program for analysis of Bayesian graphical mod-els using Gibbs sampling. In Proceedings of the 3rd International Workshopon Distributed Statistical Computing (DSC 2003), 20-22 March, Vienna,Austria. ISSN 1609-395X.

Punt, A.E., Smith, D.C. and Smith, A.D.M. 2011. Among-stock comparisonsfor improving stock assessments of data-poor stocks: the Robin Hood ap-proach. ICES J. Mar. Sci. 68(5): 972981

Rochet, M.J. 2000. A comparative approach to life-history strategies andtactics among four orders of teleost fish. ICES J. Mar. Sci. 57: 228-239.

Stan Development Team 2014. Stan: A C++ Library for Probability andSampling, Version 2.2, URL http://mc-stan.org/

Su, Z., Peterman, R.M., and Haeseker, S.L. 2004. Spatial hierarchi-cal Bayesian models for stockrecruitment analysis of pink salmon (On-corhynchus gorbuscha). Can. J. Fish. Aquat. Sci. 61(12): 2471-2486.

Uusitalo, L., Kuikka, S., and Romakkaniemi, A. 2005. Estimation of Atlanticsalmon smolt carrying capacity of rivers using expert knowledge. ICES J.Mar. Sci. 62: 708-722.

Vanpaemel, W. 2011 Constructing informative model priors using hierarchicalmethods. J. Math. Psychol. 55(1): 106-117.

Varis, O., and Kuikka, S. 1997. Joint use of multiple environmental assess-ment models by a Bayesian meta-model: The Baltic Salmon case. Ecol.Model. 102: 341-351.

Wenger, S.J. and Olden, J.D. 2012. Assessing transferability of ecologicalmodels: an underappreciated aspect of statistical validation. Methods Ecol.Evol. 3: 260-267.

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