special issue of Statistica Sinica on sequential Monte Carlo

reXing the bridge

As I was re-reading Xiao-Li  Meng’s and Wing Hung Wong’s 1996 bridge sampling paper in Statistica Sinica, I realised they were making the link with Geyer’s (1994) mythical tech report, in the sense that the iterative construction of α functions “converges to the `reverse logistic regression’  described in Geyer (1994) for the two-density cases” (p.839). Although they also saw the later as an “iterative” application of Torrie and Valleau’s (1977) “umbrella sampling” estimator. And cited Bennett (1976) in the Journal of Computational Physics [for which Elsevier still asks for $39.95!] as the originator of the formula [check (6)]. And of the optimal solution (check (8)). Bennett (1976) also mentions that the method fares poorly when the targets do not overlap:

“When the two ensembles neither overlap nor satisfy the above smoothness condition, an accurate estimate of the free energy cannot be made without gathering additional MC data from one or more intermediate ensembles”

in which case this sequence of intermediate targets could be constructed and, who knows?!, optimised. (This may be the chain solution discussed in the conclusion of the paper.) Another optimisation not considered in enough detail is the allocation of the computing time to the two densities, maybe using a bandit strategy to avoid estimating the variance of the importance weights first.

Nonparametric applications of Bayesian inference

Gary Chamberlain and Guido Imbens published this paper in the Journal of Business & Economic Statistics in 2003. I just came to read it in connection with the paper by Luke Bornn, Niel Shephard and Reza Solgi that I commented a few months ago. The setting is somewhat similar: given a finite support distribution with associated probability parameter θ, a natural prior on θ is a Dirichlet prior. This prior induces a prior on transforms of θ, whether or not they are in close form (for instance as the solution of a moment equation E[F(X,β)]=0. As in Bornn et al. In this paper, Chamberlain and Imbens argue in favour of the limiting Dirichlet with all coefficients equal to zero as a way to avoid prior dominating influence when the number of classes J goes to infinity and the data size remains fixed. But they fail to address the issue that the posterior is no longer defined since some classes get unobserved. They consider instead that the parameters corresponding to those classes are equal to zero with probability one, a convention and not a result. (The computational advantage in using the improper prior sounds at best incremental.) The notion of letting some Dirichlet hyper-parameters going to zero is somewhat foreign to a Bayesian perspective as those quantities should be either fixed or distributed according to an hyper-prior, rather than set to converge according to a certain topology that has nothing to do with prior modelling. (Another reason why setting those quantities to zero does not have the same meaning as picking a Dirac mass at zero.)

“To allow for the possibility of an improper posterior distribution…” (p.4)

This is a weird beginning of a sentence, especially when followed by a concept of expected posterior distribution, which is actually a bootstrap expectation. Not as in Bayesian bootstrap, mind. And thus this feels quite orthogonal to the Bayesian approach. I do however find most interesting this notion of constructing a true expected posterior by imposing samples that ensure properness as it reminds me of our approach to mixtures with Jean Diebolt, where (latent) allocations were prohibited to induce improper priors. The bootstrapped posterior distribution seems to be proposed mostly for assessing the impact of the prior modelling, albeit in an non-quantitative manner. (I fail to understand how the very small bootstrap sample sizes are chosen.)

Obviously, there is a massive difference between this paper and Bornn et al, where the authors use two competing priors in parallel, one on θ and one on β, which induces difficulties in setting priors since the parameter space is concentrated upon a manifold. (In which case I wonder what would happen if one implemented the preposterior idea of Berger and Pérez, 2002, to derive a fixed point solution. That we implemented recently with Diego Salmerón and Juan Antonio Caño in a paper published in Statistica Sinica.. This exhibits a similarity with the above bootstrap proposal in that the posterior gets averaged wrt another posterior.)

Objective Bayesian hypothesis testing

Our paper with Diego Salmerón and Juan Cano using integral priors for binomial regression and objective Bayesian hypothesis testing (one of my topics of interest, see yesterday’s talk!) eventually appeared in Statistica Sinica. This is Volume 25,  Number 3, of July 2015 and the table of contents shows an impressively diverse range of topics.