**A**drien Hairault (PhD student at Dauphine), Judith and I just arXived a new paper on evidence estimation for mixtures. This may sound like a well-trodden path that I have repeatedly explored in the past, but methinks that estimating the model evidence doth remain a notoriously difficult task for large sample or many component finite mixtures and even more for “infinite” mixture models corresponding to a Dirichlet process. When considering different Monte Carlo techniques advocated in the past, like Chib’s (1995) method, SMC, or bridge sampling, they exhibit a range of performances, in terms of computing time… One novel (?) approach in the paper is to write Chib’s (1995) identity for partitions rather than parameters as (a) it bypasses the label switching issue (as we already noted in Hurn et al., 2000), another one is to exploit Geyer (1991-1994) reverse logistic regression technique in the more challenging Dirichlet mixture setting, and yet another one a sequential importance sampling solution à la Kong et al. (1994), as also noticed by Carvalho et al. (2010). [We did not cover nested sampling as it quickly becomes onerous.]

Applications are numerous. In particular, testing for the number of components in a finite mixture model or against the fit of a finite mixture model for a given dataset has long been and still is an issue of much interest and diverging opinions, albeit yet missing a fully satisfactory resolution. Using a Bayes factor to find the right number of components K in a finite mixture model is known to provide a consistent procedure. We furthermore establish there the consistence of the Bayes factor when comparing a parametric family of finite mixtures against the nonparametric ‘strongly identifiable’ Dirichlet Process Mixture (DPM) model.

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