title: Connecting the latent multinomial.
abstract: 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.
authors.affiliation: Department of Mathematics and Statistics, University of Otago, P.O. Box 56, Dunedin 9054, New Zealand.
authors.forename: Matthew R
authors.initials: MR
authors.lastname: Schofield
authors.affiliation: Department of Statistics, University of Kentucky, Lexington, Kentucky 40536, U.S.A.
authors.forename: Simon J
authors.initials: SJ
authors.lastname: Bonner
mesh_terms: D000465:Algorithms; D000818:Animals; D001499:Bayes Theorem; D001699:Biometry / Q000379:methods; D016014:Linear Models; D008390:Markov Chains; D015233:Models, Statistical; D009010:Monte Carlo Method; D011157:Population Dynamics / Q000706:statistics & numerical data*
keywords: Capture-recapture; Linear constraint; Markov basis; Markov chain Monte Carlo; Misidentification