New Pub: A full-capture Hierarchical Bayesian model of Pollock’s Closed Robust Design and application to dolphins

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Rankin, R. W., Nicholson, K. E., Allen, S. J., Krützen, M., Bejder, L., & Pollock, K. H. (2016). A full-capture Hierarchical Bayesian model of Pollock’s Closed Robust Design and application to dolphins. Frontiers in Marine Science, 3(25). doi:10.3389/fmars.2016.00025

We are pleased to announce our new publication on Shark Bay bottlenose dolphins which benchmarks model-averaging in Program MARK and a Bayesian Hierarchical model for temporary-migration Robust Design mark-recapture models. See below for the citation, link to the free full-text PDF, and an online R/JAGS tutorial for Bayesian mark-capture.

Alternative to AIC Model-Averaging

The paper will be of interest to cetacean researchers who use Program MARK for temporary-migration Robust Design models. In particular that we show that a Hierarchical Bayesian model can yield similar estimates as model-averaging by AICc, the latter being the current best-practise to deal with the vast number of ‘fixed-effects’ models that one typically considers. Model-averaging and Bayesian frameworks have some similar philosophical underpinnings, such as conditioning on the data (Burnham and Anderson 2014). However, the HB framework is also highly extensible and can deal with other challenges where the AIC is undefined, such as random-effects and individual-level heterogeneity in capture-probabilities.

Mark-Recapture and low-sample sizes: the Bayesian Answer

Bayesian models are a solid answer to a perennial dilemma among cetacean researchers: photo-ID datasets are typically sparse or have low-sample sizes. In contrast, researchers typically want complex data-hungry model to increase ecological realism. For example, a simple temporary-migration model or individual heterogeneity model will demand >30 – 70 variables for a mid-sized dataset. Frequentist and AICc-based inference will be overly-confident in such situations, and yield ridiculous estimates such as 100% detection, or 0% migration, or 100% survival, or just fail altogether. Alternatively, Hierarchical Bayesian models provide exact inference under low-sample sizes: they just depend more on the prior distributions, which, if set-up thoughtfully, are more conservative, make better predictions, and can automatically safeguard against over-parametrization (Berger 2006, Gelman 2013).

Individual Heterogeneity

As the distribution of individual capture probabilities gets wider, the least-captureable individuals are more likely to be missed altogether (grey area), leading to a  biased high estimate of the mean capture parameter (p-hat), and biased low population abundance estimates (top-right corner)

As the distribution of individual capture probabilities gets wider, the least-captureable individuals are more likely to be missed altogether (grey area), leading to a biased high estimate of the mean capture parameter (p-hat), and biased low population abundance estimates (top-right corner)

Individual heterogeneity in capture probabilities will result in biased-low population abundance estimates (see animation), and therefore it is a primary preoccupation of most capture-recapture practitioners. Under a Hierarchical Bayesian full-capture framework, it is trivial to model individuals as coming from a distribution, without a large increase in complexity. In contrast, the comparable fixed-effect version in Program MARK, the ‘two-point finite mixture model’, typically yields over-parametrized models and unreliable capture-estimates (e.g., p=1).

R and JAGS code

See our online R/JAGS tutorial at Github for code to run the Hierarchical Bayesian Pollock’s Closed Robust Design. The tutorial includes an example photo-ID bottlenose dolphin dataset from Krista et al. 2012. We use the flexible BUGS-like Bayesian syntax called “JAGS”, which makes Bayesian models accessible to almost anyone with rudimentary scripting skills.

Key Findings

  • full-capture, non-hierarchical Bayesian PCRD models had slightly better estimation performance than equivalent fixed-effects Maximum-Likelihood estimation (in MARK), mainly due to the latter’s susceptibility to singularities (although there was no clear champion);
  • we propose a Hierarchical Bayesian PCRD which can lead to similar estimates as AICc model-averaging and serve as a type of multi-model inference;
  • we showed how heterogeneity in detection probabilities can lead to a 8-24% increase in bottlenose dolphin abundance estimates, as compared to ML and Bayesian models that assume homogeneous detection probabilities;
  • we explored the partial non-identifiability and high correlation among parameter estimates, especially between survival and temporary-migration which has serious consequences for ones’ ability to use these parameters for inference, and which should influence researchers’ study design and modelling strategies;
  • we proposed two posterior predictive checks to help diagnose poor model fitting, in lieu of a formal goodness-of-fit procedure in popular CMR software.

  • But Aren’t Bayesian’s Biased?

    Some Mark users who are new to Bayesian inference may worry about prior information and the inherent bias of subjective Bayesian models. A subjective Prior contains information. This information competes with the information in the data to influence the Posterior distribution (the probability distribution which is used to make conclusions about a variable of interesheterogenietyt, like population abundance or survival). The more information in a Prior, the more a posterior expectation will be pulled away from the “objective” MLE and towards the prior expectation; conversely, the more information in our data (e.g., a large sample size), the prior becomes less influential and inferences are practically the same between ML-based and Bayesian paradigms.

    But, there is strong evidence from the machine-learning and predictive analytics community that slightly conservatively biased models yield better predictions, especially in the face of low-sample sizes and very complex models (Murphy KP, 2012). In the Learning community, this is called “Regularization“, such as the Lasso or Ridge Regression or Boosting: these techniques impose a penalty on model complexity and favour simpler models than “objective” ML models estimate. Interestedly, many of the Learning communities’ regularization techniques are actually a type of Bayesian model (Hooten and Hobbs 2015).

    The phenomenon is nicely illustrated with a coin toss: Imagine you want to estimate the fairness of a coin (H vs T), but you flip it only 3 times. Let’s say you observed TTT (a rare, but not impossible event). The Maximum Likelihood Estimate would be that the probability of a tail is 100%, and the coin is definitely not fair. Of course, we know this is a terrible estimate and a stupid example, because we have solid prior information about the behaviour of coins. However, the sad fact is that many Dolphin Mark-Recapture researchers are making similarly stupid experiments, under small sample sizes and overly complex Frequentist models which estimate of 100% detection probability or 100% survival or 0% movement.

    In contrast, a little bit of skepticism encoded in a Bayesian prior (like Beta(1,1)) can temper our estimates. For our coin example, the posterior would say that the probability of a tail is just 80%, given the observed sequence TTT. That’s still not the true 0.5, but it is a better prediction than the frequentist’s 100%

    Abstract

    We present a Hierarchical Bayesian version of Pollock's Closed Robust Design for studying the survival, temporary-migration, and abundance of marked animals. Through simulations and analyses of a bottlenose dolphin photo-identification dataset, we compare several estimation frameworks, including Maximum Likelihood estimation (ML), model-averaging by AICc, as well as Bayesian and Hierarchical Bayesian (HB) procedures. Our results demonstrate a number of advantages of the Bayesian framework over other popular methods. First, for simple fixed-effect models, we show the near-equivalence of Bayesian and ML point-estimates and confidence/credibility intervals. Second, we demonstrate how there is an inherent correlation among temporary-migration and survival parameter estimates in the PCRD, and while this can lead to serious convergence issues and singularities among MLEs, we show that the Bayesian estimates were more reliable. Third, we demonstrate that a Hierarchical Bayesian model with carefully thought-out hyperpriors, can lead to similar parameter estimates and conclusions as multi-model inference by AICc model-averaging. This latter point is especially interesting for mark-recapture practitioners, for whom model-uncertainty and multi-model inference have become a major preoccupation. Lastly, we extend the Hierarchical Bayesian PCRD to include full-capture histories (i.e., by modelling a recruitment process) and individual-level heterogeneity in detection probabilities, which can have important consequences for the range of phenomena studied by the PCRD, as well as lead to large differences in abundance estimates. For example, we estimate 8%-24% more bottlenose dolphins in the western gulf of Shark Bay than previously estimated by ML and AICc-based model-averaging. Other important extensions are discussed. Our Bayesian PCRD models are written in the BUGS-like JAGS language for easy dissemination and customization by the community of capture-mark-recapture practitioners.
    

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    Copy the following Bibtex into your favourite Reference Manager.
    @article{rankin_full-capture_2016,
    title = {A full-capture {Hierarchical} {Bayesian} model of {Pollock}'s {Closed} {Robust} {Design} and application to dolphins},
    volume = {3},
    url = {http://journal.frontiersin.org/article/10.3389/fmars.2016.00025},
    doi = {10.3389/fmars.2016.00025},
    abstract = {We present a Hierarchical Bayesian version of Pollock's Closed Robust Design for studying the survival, temporary migration, and abundance of marked animals. Through simulations and analyses of a bottlenose dolphin photo-identification dataset, we compare several estimation frameworks, including Maximum Likelihood estimation (ML), model-averaging by AICc, as well as Bayesian and Hierarchical Bayesian (HB) procedures. Our results demonstrate a number of advantages of the Bayesian framework over other popular methods. First, for simple fixed-effect models, we show the near-equivalence of Bayesian and ML point-estimates and confidence/credibility intervals. Second, we demonstrate how there is an inherent correlation among temporary migration and survival parameter estimates in the PCRD, and while this can lead to serious convergence issues and singularities among MLEs, we show that the Bayesian estimates were more reliable. Third, we demonstrate that a Hierarchical Bayesian model with carefully thought-out hyperpriors, can lead to similar parameter estimates and conclusions as multi-model inference by AICc model-averaging. This latter point is especially interesting for mark-recapture practitioners, for whom model-uncertainty and multi-model inference have become a major preoccupation. Lastly, we extend the Hierarchical Bayesian PCRD to include full-capture histories (i.e., by modelling a recruitment process) and individual-level heterogeneity in detection probabilities, which can have important consequences for the range of phenomena studied by the PCRD, as well as lead to large differences in abundance estimates. For example, we estimate 8\%-24\% more bottlenose dolphins in the western gulf of Shark Bay than previously estimated by ML and AICc-based model-averaging. Other important extensions are discussed. Our Bayesian PCRD models are written in the BUGS-like JAGS language for easy dissemination and customization by the community of capture-mark-recapture practitioners.},
    number = {25},
    journal = {Frontiers in Marine Science},
    author = {Rankin, Robert W. and Nicholson, Krista E. and Allen, Simon J. and Krützen, Michael and Bejder, Lars and Pollock, Kenneth H.},
    year = {2016}
    }

    Originally posted here

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