ghost [parameters] in the [Bayesian] shell

This question appeared on Stack Exchange (X Validated) two days ago. And the equalities indeed seem to suffer from several mathematical inconsistencies, as I pointed out in my Answer. However, what I find most crucial in this question is that the quantity on the left hand side is meaningless. Parameters for different models only make sense within their own model. Hence when comparing models parameters cannot co-exist across models. What I suspect [without direct access to Kruschke’s Doing Bayesian Data Analysis book and as was later confirmed by John] is that he is using pseudo-priors in order to apply Carlin and Chib (1995) resolution [by saturation of the parameter space] of simulating over a trans-dimensional space…

Savage-Dickey supermodels

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,

B¹⁰=π(φ⁰|x)/π(φ⁰).

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].

a day for comments

As I was flying over Skye (with [maybe] a first if hazy perspective on the Cuillin ridge!) to Iceland, three long sets of replies to some of my posts appeared on the ‘Og:

• Dan Simpson replied to my comments of last Tuesday about his PC  construction;
• Arnaud Doucet precised some issues about his adaptive subsampling paper;
• Amandine Schreck clarified why I had missed some points in her Bayesian variable selection paper;
• Randal Douc defended the efficiency of using Carlin and Chib (1995) method for mixture simulation.

Thanks to them for taking the time to answer my musings…

 

Carlin and Chib (1995) for fixed dimension problems

Yesterday, I was part of a (public) thesis committee at the Université Pierre et Marie Curie, in down-town Paris. After a bit of a search for the defence room (as the campus is still undergoing a massive asbestos clean-up, 20 years after it started…!), I listened to Florian Maire delivering his talk on an array of work in computational statistics ranging from the theoretical (Peskun ordering) to the methodological (Monte Carlo online EM) to the applied (unsupervised learning of classes shapes via deformable templates). The implementation of the online EM algorithm involved the use of pseudo-priors à la Carlin and Chib (1995), even though the setting was a fixed-dimension one, in order to fight the difficulty of exploring the space of templates by a regular Gibbs sampler. (As usual, the design of the pseudo-priors was crucial to the success of the method.) The thesis also included a recent work with Randal Douc and Jimmy Olsson on ranking inhomogeneous Markov kernels of the type

against alternatives with components (P’,Q’). The authors were able to characterise minimal conditions for a Peskun-ordering domination on the components to transfer to the combination. Quite an interesting piece of work for a PhD thesis!

Death sequence

August is not looking kindly at statisticians as I have now learned (after ten days of disconnection) of both Arnold Zellner and John Nelder passing away, on Aug. 11 and 15, respectively. Following this close the death of Julian Besag, this is a sad series of departures of leading figures in the fields of statistics and econometrics. Arnold was 83 and, although I had met him in several Valencia meetings—including one in Alicante where we sat together for breakfast with Persi Diaconis and  where an irate [and well-known ] statistician came to Arnold demanding apologies about comments made late the night before!—, I only had true interactions with him during the past years, over the Jeffreys reassessment I conducted with Judith Rousseau and Nicolas Chopin. On this occasion, Arnold was very kindly helpful, pointing out the volume that he had edited on Jeffreys and that I overlooked, discussing more philosophical points about the early part of Theory of Probability, and making a very nice overview of it at the O’Bayes 09 meeting. Always in the kindest manner. Sid Chib wrote an obituary of Arnold Zellner on the ISBA website (Arnold was the first ISBA president). Andrew Gelman also wrote some personal recollections about Arnold. A memorial site has been set up in his honour.

John Nelder was regularly attending the Read Paper sessions at the RSS and these are the only times I met him. He was an impressive figure in many ways, first and foremost for his monumental Generalised Linear Models with Peter McCullagh, a (difficult and uncompromising) book that I strongly recommend to (i.e. force upon!) my PhD students for its depth. I also remember being quite intimidated the first time I talked with him, failing to understand his arguments so completely that I dreaded later discussions… John Nelder was at  Fisher’s Rothamsted Experimental Station for most of his career and was certainly one of the last genuine Fisherians (despite a fairly rude letter of Fisher to him!).