Here p ( · ) a prior distribution for parameter vector θ , which may itself have one or more levels of stochastic interdependence. The distribution of each HMM initial state x$_{i}$$_{1}$ is f$_{i}$$_{1}$ ( · | θ ). Markov state transition probabilities are given by f$_{it}$ ( · | θ, x$_{i,t}$$_{−}$$_{1}$ ) and observation probabilities by g$_{it}$ ( · | θ, x$_{it}$ ). Discrete HMMs have long been applied in the area of ecological capture-recapture ( e.g. , Gimenez et al., 2007; King, 2012; Langrock et al., 2012). In this context, a set of n distinct animals is monitored for k sampling occasions. Each y$_{i}$ represents the observation history of animal i , for i = 1 , . . . , n , which can be modeled using HMMs as in (1). The set of possible observations Y may include a state to represent “unobserved”. Since all n animals are not typically observed on occasion t = 1, each embedded HMM will “begin” at the sampling period corresponding to the first genuine observation of that animal. We now provide the model likelihood for the general HMM formulation in (1), which is used in the Bayesian estimation procedures described next. We begin with the likelihood contribution from a single observation history, where X k denotes the standard k -fold Cartesian product of X . Using the likelihood compo- nents in (2), the total model likelihood of y is We now describe several computational approaches to applying Bayesian estimation to em- bedded HMMs. These strategies will form the basis for our comparisons, using examples 6 2.3 Computational Approaches 2.2 Model Likelihood L ( θ | y$_{i}$ ) = ∑ x$_{i}$ ∈X k f$_{i}$$_{1}$ ( x$_{i}$$_{1}$ | θ )$^{(}$ k ∏ t =2 f$_{it}$ ( x$_{it}$ | θ, x$_{i,t}$$_{−}$$_{1}$ )$^{) (}$ k ∏ t =1 g$_{it}$ ( y$_{it}$ | θ, x$_{it}$ )$^{)}$ , (2) L ( θ | y ) = n ∏ i =1 L ( θ | y$_{i}$ ) .