In connection with the Valencia 9 meeting that started yesterday, and with Hedie‘s talk there, we have posted on arXiv a set of comments on particle learning. The arXiv paper contains several discussions but they mostly focus on the inevitable degeneracy that accompanies particle systems. When Lopes et al. state that is not of interest as the filtered, low dimensional is sufficient for inference at time t, they seem to implicitly imply that the restriction of the simulation focus to a low dimensional vector is a way to avoid the degeneracy inherent to all particle filters. The particle learning algorithm therefore relies on an approximation of and the fact that this approximation quickly degenerates as t increases means that this approximation impacts the approximation of . We show that, unless the size of the particle population exponentially increases with t, the sample of ‘s will not be distributed as an iid sample from .
The graph above is an illustration of the degeneracy in the setup of a Poisson mixture with five components and 10,000 observations. The boxplots represent the variation of the evidence approximations based on a particle learning sample and Lopes et al. approximation, on a particle learning sample and Chib’s (1995) approximation, and on an MCMC sample and Chib’s (1995) approximation, for 250 replications. The differences are therefore quite severe when considering this number of observations. (I put the R code on my website for anyone who wants to check if I programmed things wrong.) There is no clear solution to the degeneracy problem, in my opinion, because the increase in the particle size overcoming degeneracy must be particularly high… We will be discussing that this morning.