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Sampling from the conditional (or posterior) probability distribution of thelatent states of a Hidden Markov Model, given the realization of the observedprocess, is a non-trivial problem in the context of Markov Chain Monte Carlo.To do this Andrieu et al. (2010) constructed a Markov kernel which leaves thisconditional distribution invariant using a Particle Filter. From apractitioner's point of view, this Markov kernel attempts to mimic the act ofsampling all the latent state variables as one block from the posteriordistribution but for models where exact simulation is not possible. There aresome recent theoretical results that establish the uniform ergodicity of thisMarkov kernel and that the mixing rate does not diminish provided the number ofparticles grows at least linearly with the number of latent states in theposterior. This gives rise to a cost, per application of the kernel, that isquadratic in the number of latent states which could be prohibitive for longobservation sequences. We seek to answer an obvious but important question: isthere a different implementation with a cost per-iteration that grows linearlywith the number of latent states, but which is still stable in the sense thatits mixing rate does not deteriorate? We address this problem using blockingstrategies, which are easily parallelizable, and prove stability of theresulting sampler.

Blocking Strategies and Stability of Particle Gibbs Samplers