Reviews: Parameter elimination in particle Gibbs sampling

The marginalisation of variables within some steps of an MCMC algorithm is delicate. The main proposal here appears well justified, but it would have been nice to see the argument made a little more explicitly. The type of marginalisation described here seems to be more or less what would be described as a (partially) collapsed Gibbs sampler in the sense of [David A Van Dyk and Taeyoung Park. "Partially collapsed Gibbs samplers: Theory and methods". It was less clear to me exactly how the "blocking" strategy detailed in Section 4.1 would be justified from a formal perspective, and I do think that this needs clarifying. I.e. the collection of variables to be sampled is divided into three parts -- x', x and theta and the decomposition of the kernel seems to involve sampling: x from a kernel invariant to its distribution conditional on both x' and theta (starting from the previous x) x' from a kernel invariant with respect to its distribution conditional only upon x (starting from the previous x') \theta from its full conditional distribution and it's not completely transparent how one knows that this is invariant with respect to the correct joint distribution.