[Paper Review] Examining posterior propriety in the Bayesian analysis of capture-recapture models

Arjun M. Gopalaswamy, Mohan Delampady|arXiv (Cornell University)|Nov 8, 2016

Census and Population Estimation14 references3 citations

TL;DR

This paper challenges the widespread adoption of the scale prior in capture-recapture models, arguing that posterior impropriety—previously attributed to the discrete uniform prior—is rarely an issue when using efficient likelihoods like $M_0$ and $M_h$. By analyzing the asymptotic behavior of posterior distributions using Big-Oh notation, the authors derive new propriety theorems, showing that proper posteriors are generally ensured with realistic ecological likelihoods, and caution against blind application of mathematical results without domain-specific validation.

ABSTRACT

There lies a latent danger in utilizing some known mathematical results in ecology. Some results do not apply to the problem at hand. We identify one such trend. Based on a couple of theorems in mathematical statistics, Link (2013) cautions ecologists about the inappropriateness of using the discrete uniform prior in their analysis under certain conditions and instead recommends the routine use of the scale prior during analysis. This recommendation is been absorbed immediately and widely among ecologists. In this study, we consider the two fundamental capture-recapture models used widely in ecology, $M_0$ and $M_h$, and derive conditions for posterior propriety by examining the behavior of the right tail of the posterior distributions of animal population size $N$ in a Bayesian analysis. We demonstrate that both these likelihoods are far more efficient than the ones considered in Link (2013). We argue that no particularly prescriptive approach should be adopted by ecologists in regard to choosing priors of the fear of posterior impropriety. Instead, we recommend the efficient construction of likelihoods for the problem and data on hand, choosing priors based existing knowledge of a parameter of interest and encourage examining posterior propriety by asymptotic arguments as demonstrated in this study.

Motivation & Objective

• To investigate whether the posterior propriety concerns raised by Link (2013) apply to widely used capture-recapture models $M_0$ and $M_h$.
• To assess the relevance of mathematical results from Kahn (1987) and York & Madigan (1992) to real ecological likelihoods in animal abundance estimation.
• To develop new propriety theorems for $M_0$ and $M_h$ by analyzing the asymptotic behavior of the posterior tail as $N \to \infty$.
• To argue against the routine adoption of the scale prior and advocate for likelihood-informed, context-specific prior selection.
• To emphasize the importance of examining posterior propriety through asymptotic analysis rather than relying on generalized mathematical results.

Proposed method

• Derives conditions for posterior propriety by examining the right-tail behavior of the posterior distribution of $N$ as $N \to \infty$.
• Applies Big-Oh notation (from complexity theory) to characterize the asymptotic decay rate of the likelihood function in $M_0$ and $M_h$ models.
• Compares the likelihood structures of $M_0$ and $M_h$ with those in Kahn (1987) and York & Madigan (1992), highlighting structural inefficiencies in the latter.
• Uses asymptotic arguments to demonstrate that the $M_0$ and $M_h$ likelihoods are far more efficient than those in the foundational papers, reducing the risk of impropriety.
• Re-evaluates the claim of 'Bayesian stupefaction' in Link (2013) by analyzing the impact of data augmentation parameter $M$ on posterior standard deviation and mean.
• Advocates for a shift from prescriptive prior rules (e.g., scale prior) to model-specific likelihood construction and posterior propriety checks via asymptotic analysis.

Experimental results

Research questions

• RQ1 Does the discrete uniform prior lead to improper posteriors in the $M_0$ and $M_h$ capture-recapture models under realistic ecological conditions?
• RQ2 How do the likelihood structures of $M_0$ and $M_h$ compare to those in Kahn (1987) and York & Madigan (1992), which underpin Link (2013)'s recommendations?
• RQ3 Can posterior propriety be rigorously assessed through asymptotic analysis of the likelihood's tail behavior?
• RQ4 Why does the scale prior recommendation from Link (2013) lead to misleading inferences in some MCMC implementations, and is this due to prior choice or likelihood inefficiency?
• RQ5 To what extent is the phenomenon of 'Bayesian stupefaction' a consequence of inefficient likelihoods rather than improper priors?

Key findings

• The $M_0$ and $M_h$ likelihoods are asymptotically more efficient than those in Kahn (1987) and York & Madigan (1992), reducing the risk of posterior impropriety.
• Posterior propriety for $M_0$ and $M_h$ is established through asymptotic analysis using Big-Oh notation, showing that the posterior tail decays sufficiently fast to ensure integrability.
• The observed reduction in posterior standard deviation when increasing $M$ in Link (2013)'s analysis is likely due to the scale prior's concentration at $\psi \approx 0.1$, not posterior impropriety.
• The discrete uniform prior does not lead to improper posteriors in $M_0$ and $M_h$ under standard ecological assumptions, contradicting Link (2013)'s generalization.
• The phenomenon of 'Bayesian stupefaction' is better explained by likelihood inefficiency than by prior choice, as the likelihoods in Kahn and York & Madigan are poorly suited to capture-recapture data.
• Ecologists should prioritize constructing efficient likelihoods based on sampling mechanisms and assess propriety via asymptotic analysis, rather than defaulting to scale priors.