Connecting the latent multinomial

• Published in: Biometrics 2015-12, Vol.71 (4), p.1070-1080
• Main Authors: Schofield, Matthew R, Bonner, Simon J
• Format: Article
• Language: English
• Subjects: Algorithms; Animals; Bayes Theorem; Bayesian analysis; biometry; Biometry - methods; Capture-recapture; DISCUSSION PAPER; Linear constraint; Linear Models; Markov analysis; Markov basis; Markov chain; Markov chain Monte Carlo; Markov Chains; Misidentification; Models, Statistical; Monte Carlo Method; Population Dynamics - statistics & numerical data
• ISSN: 0006-341X; 1541-0420
• DOI: 10.1111/biom.12333

Link et al. (2010, Biometrics 66, 178–185) 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=Ax, with x assumed to follow a multinomial distribution conditional on the model parameters, θ. Bayesian methods are then applied by sampling from the joint posterior distribution of both x and θ. 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, The Annals of Statistics 26, 363–397). 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 in 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.