Model choice by Kullback projection (2)

Yesterday I talked about the paper of Nott and Cheng at the Bayesian model choice group and [with the help of the group] realised that my earlier comment on the paper

There is however one point with which I disagree, namely that the predictive on the submodel is obtained in the current paper by projecting a Monte Carlo or an MCMC sample from the predictive on the full model, while I think this is incorrect because the likelihood is then computed using the parameter for the full model. Using a projection of such a sample means at least reweighting by the ratio of the likelihoods…

was not completely accurate. The point is [I think] correct when considering the posterior distribution of the projected parameters. Thus, using a projection of an MCMC sample corresponing to the full model will not result in a sample from the posterior distribution of the projected parameters. On the other hand, projecting the MCMC sample in order to get the Kullback-Leibler distance posterior distribution as done in the applications of Section 7 of the paper is completely kosher, since this is a quantity that only depends on the full model parameters. Since Nott and Cheng do not consider the projected model at any time (even though Section 3 is slightly unclear, using a posterior on the projected parameter), there is nothing wrong in their paper and I do find quite interesting the idea that the lasso penalty allows for a simultaneous exploration of the most likely submodels without a recourse to a more advanced technique like reversible jump. (The comparison is obviously biased as the method does not provide a true posterior on the most likely submodels, only an approximation of their probability. Simulating from the constrained projected posterior would require extra steps.)