from capture-recapture. One approach to Bayesian estimation is to perform MCMC sampling of both the model parameters and latent states; that is, to sample from the full posterior distribution p ( θ, x | y ). Doing so makes use of Bayes law in the form: Using this approach the dimension of the MCMC sampling problem can be very large, since there can be up to nk latent state variables. Although we expect the MCMC update of each individual variable will be fast, since the algorithmic complexity is limited to that of standard MCMC sampling algorithms ( e.g. , Metropolis-Hastings), there can be a large number of latent states. In addition to the computational cost, this can result in slow MCMC mixing for latent states and parameters. An alternate approach makes use of direct filtering to calculate the likelihood contribution of each observation history. This approach relies on the discrete HMM structure underlying each observed sequence y$_{i}$ in (1). Doing so, we may perform MCMC sampling of the posterior distribution of θ only, rather than ( θ, x ) as in the latent state MCMC, and use filtering to calculate each p ( y$_{i}$ | θ ) as described in Elliott, Aggoun, and Moore (2008). The filtering MCMC approach makes use of Bayes law in the form: For a general discrete HMM as specified in (1), the filtering likelihood calculation proceeds as follows. Everything pertains to the i th observation history y$_{i}$ and we omit subscripts i . 7 p ( θ | y ) ∝ p ( θ ) n ∏ i =1 p ( y$_{i}$ | θ ) (3) Filtering over latent states with MCMC for parameters p ( θ, x | y ) ∝ p ( θ ) n ∏ i =1 p ( x$_{i}$ | θ ) p ( y$_{i}$ | θ, x$_{i}$ ) MCMC for latent states and parameters