Connecting the latent multinomial

TL;DR

This paper refines Bayesian methods for capture-recapture data analysis by emphasizing the necessity of Markov bases over simple bases for ensuring irreducible Markov chains, with proofs and examples illustrating their importance.

Contribution

It demonstrates that using Markov bases is essential for proper Bayesian sampling in capture-recapture models and provides a constructive method to obtain such bases.

Findings

Markov bases ensure irreducibility in Markov chain sampling

A specific lattice basis is proven to be a Markov basis for certain models

The paper offers a method to construct Markov bases for classes of models

Abstract

Link et al. (2010) define a general framework for analyzing capture-recapture data with potential misidentifications. In this framework, the observed vector of counts, $y$, is considered as a linear function of a vector of latent counts, $x$, such that $y = A x$, with $x$ assumed to follow a multinomial distribution conditional on the model parameters, $\theta$. Bayesian methods are then applied by sampling from the joint posterior distribution of both $x$ and $\theta$. In particular, Link et al. (2010) propose a Metropolis-Hastings algorithm to sample from the full conditional distribution of $x$, where new proposals are generated by sequentially adding elements from a basis of the null space (kernel) of $A$. We consider this algorithm and show that using elements from a simple basis for the kernel of $A$ may not produce an irreducible Markov chain. Instead, we require a Markov basis, as defined by Diaconis and Sturmfels (1998). We illustrate the importance of Markov bases with three capture-recapture examples. We prove that a specific lattice basis is a Markov basis for a class of models including the original model considered by Link et al. (2010) and confirm that the specific basis used by Link et al. (2010) for their example with two sampling occasions is a Markov basis. The constructive nature of our proof provides an immediate method to obtain a Markov basis for any model in this class.