A. Mootoovaloo, B. Bassett, and M. Kunz just arXived a paper on the computation of Bayes factors by the Savage-Dickey representation through a supermodel (or encompassing model). (I wonder why Savage-Dickey is so popular in astronomy and cosmology statistical papers and not so much elsewhere.) Recall that the trick is to write the Bayes factor in favour of the encompasssing model as the ratio of the posterior and of the prior for the tested parameter (thus eliminating nuisance or common parameters) at its null value,
Modulo some continuity constraints on the prior density, and the assumption that the conditional prior on nuisance parameter is the same under the null model and the encompassing model [given the null value φ⁰]. If this sounds confusing or even shocking from a mathematical perspective, check the numerous previous entries on this topic on the ‘Og!
The supermodel created by the authors is a mixture of the original models, as in our paper, and… hold the presses!, it is a mixture of the likelihood functions, as in Phil O’Neill’s and Theodore Kypraios’ paper. Which is not mentioned in the current paper and should obviously be. In the current representation, the posterior distribution on the mixture weight α is a linear function of α involving both evidences, α(m¹-m²)+m², times the artificial prior on α. The resulting estimator of the Bayes factor thus shares features with bridge sampling, reversible jump, and the importance sampling version of nested sampling we developed in our Biometrika paper. In addition to O’Neill and Kypraios’s solution.
The following quote is inaccurate since the MCMC algorithm needs simulating the parameters of the compared models in realistic settings, hence representing the multidimensional integrals by Monte Carlo versions.
“Though we have a clever way of avoiding multidimensional integrals to calculate the Bayesian Evidence, this new method requires very efficient sampling and for a small number of dimensions is not faster than individual nested sampling runs.”
I actually wonder at the sheer rationale of running an intensive MCMC sampler in such a setting, when the weight α is completely artificial. It is only used to jump from one model to the next, which sound quite inefficient when compared with simulating from both models separately and independently. This approach can also be seen as a special case of Carlin’s and Chib’s (1995) alternative to reversible jump. Using instead the Savage-Dickey representation is of course infeasible. Which makes the overall reference to this method rather inappropriate in my opinion. Further, the examples processed in the paper all involve (natural) embedded models where the original Savage-Dickey approach applies. Creating an additional model to apply a pseudo-Savage-Dickey representation does not sound very compelling…
Incidentally, the paper also includes a discussion of a weird notion, the likelihood of the Bayes factor, B¹², which is plotted as a distribution in B¹², most strangely. The only other place I met this notion is in Murray Aitkin’s book. Something’s unclear there or in my head!
“One of the fundamental choices when using the supermodel approach is how to deal with common parameters to the two models.”
This is an interesting question, although maybe not so relevant for the Bayes factor issue where it should not matter. However, as in our paper, multiplying the number of parameters in the encompassing model may hinder convergence of the MCMC chain or reduce the precision of the approximation of the Bayes factor. Again, from a Bayes factor perspective, this does not matter [while it does in our perspective].