## Sampling latent states for high-dimensional non-linear state space models with the embedded HMM method

**P**reviously, I posted a comment on a paper by Alex Shestopaloff and Radford Neal, after my visit to Toronto two years ago, using a particular version of ensemble Monte Carlo. A new paper by the same authors was recently arXived, as an refinement of the embedded HMM paper of Neal (2003), in that the authors propose a new and more efficient way to generate from the (artificial) embedded hidden Markov sampler that is central to their technique of propagating a set of pool states. The method exploits both forward and backward representations of HMMs in an alternating manner. And propagates the pool states from one observation time to the next. The paper also exploits latent Gaussian structures to make autoregressive proposals, as well as flip proposals from x to -x [which seem to only make sense when 0 is a central value for the target, i.e. when the observables y only depend on |x|]. All those modifications bring the proposal quite close to (backward) particle Gibbs, the difference being in using Metropolis rather than importance steps. And in an improvement brought by the embedded HMM approach, even though it is always delicate to generalise those comparisons when some amount of calibration is required by both algorithms under comparison. (Especially delicate when it is rather remote from my area of expertise!) Anyway, I am still intrigued [in a positive way] by the embedded HMM idea as it remains mysterious that a finite length HMM simulation can improve the convergence performances that much. And wonder at a potential connection with an earlier paper of Anthony Lee and Krys Latuszynski using a random number of auxiliary variables. Presumably a wrong impression from a superficial memory…

March 17, 2016 at 2:58 pm

Hi Christian, thank you for reading our paper! One thing we want to point out is that flip updates are just one example of how specialized MCMC updates to generate pool states can be included when the model has a particular structure. For other situations, we would need to design other MCMC updates to generate pool states, which we hope is easier to do than designing importance densities, especially when the state space is high-dimensional.