Connecting the latent multinomial. Issue 4 (1st June 2015) - Schofield, Matthew R.; Bonner, Simon J.
Summary: 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 = A x, 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.