using mixtures towards Bayes factor approximation
Phil O’Neill and Theodore Kypraios from the University of Nottingham have arXived last week a paper on “Bayesian model choice via mixture distributions with application to epidemics and population process models”. Since we discussed this paper during my visit there earlier this year, I was definitely looking forward the completed version of their work. Especially because there are some superficial similarities with our most recent work on… Bayesian model choice via mixtures! (To the point that I misunderstood at the beginning their proposal for ours…)
The central idea in the paper is that, by considering the mixture likelihood
where x corresponds to the entire sample, it is straighforward to relate the moments of α with the Bayes factor, namely
which means that estimating the mixture weight α by MCMC is equivalent to estimating the Bayes factor.
What puzzled me at first was that the mixture weight is in fine estimated with a single “datapoint”, made of the entire sample. So the posterior distribution on α is hardly different from the prior, since it solely varies by one unit! But I came to realise that this is a numerical tool and that the estimator of α is not meaningful from a statistical viewpoint (thus differing completely from our perspective). This explains why the Beta prior on α can be freely chosen so that the mixing and stability of the Markov chain is improved: This parameter is solely an algorithmic entity.
There are similarities between this approach and the pseudo-prior encompassing perspective of Carlin and Chib (1995), even though the current version does not require pseudo-priors, using true priors instead. But thinking of weakly informative priors and of the MCMC consequence (see below) leads me to wonder if pseudo-priors would not help in this setting…
Another aspect of the paper that still puzzles me is that the MCMC algorithm mixes at all: indeed, depending on the value of the binary latent variable z, one of the two parameters is updated from the true posterior while the other is updated from the prior. It thus seems unlikely that the value of z would change quickly. Creating a huge imbalance in the prior can counteract this difference, but the same problem occurs once z has moved from 0 to 1 or from 1 to 0. It seems to me that resorting to a common parameter [if possible] and using as a proposal the model-based posteriors for both parameters is the only way out of this conundrum. (We do certainly insist on this common parametrisation in our approach as it is paramount to the use of improper priors.)
“In contrast, we consider the case where there is only one datum.”
The idea in the paper is therefore fully computational and relates to other linkage methods that create bridges between two models. It differs from our new notion of Bayesian testing in that we consider estimating the mixture between the two models in comparison, hence considering instead the mixture
which is another model altogether and does not recover the original Bayes factor (Bayes factor that we altogether dismiss in favour of the posterior median of α and its entire distribution).