Following the posting on arXiv of the Statistical Science paper of Carvalho et al., and the publication by the same authors in Bayesian Analysis of Particle Learning for general mixtures I noticed on Hedibert Lopes’ website his rejoinder to the discussion of his Valencia 9 paper has been posted. Since the discussion involved several points made by members of the CREST statistics lab (and covered the mixture paper as much as the Valencia 9 paper), I was quite eager to read Hedie’s reply. Unsurprisingly, this rejoinder is however unlikely to modify my reservations about particle learning. The following is a detailed examination of the arguments found in the rejoinder but requires a preliminary reading of the above papers as well as our discussion.. Continue reading
Archive for degeneracy
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.