  We propose a new scheme for selecting pool states for the embedded Hidden
Markov Model (HMM) Markov Chain Monte Carlo (MCMC) method. This new scheme
allows the embedded HMM method to be used for efficient sampling in state space
models where the state can be high-dimensional. Previously, embedded HMM
methods were only applied to models with a one-dimensional state space. We
demonstrate that using our proposed pool state selection scheme, an embedded
HMM sampler can have similar performance to a well-tuned sampler that uses a
combination of Particle Gibbs with Backward Sampling (PGBS) and Metropolis
updates. The scaling to higher dimensions is made possible by selecting pool
states locally near the current value of the state sequence. The proposed pool
state selection scheme also allows each iteration of the embedded HMM sampler
to take time linear in the number of the pool states, as opposed to quadratic
as in the original embedded HMM sampler. We also consider a model with a
multimodal posterior, and show how a technique we term "mirroring" can be used
to efficiently move between the modes.
