Archive for Zurich
At last, we have completed, arXived, and submitted our paper on the evaluation of summary statistics for Bayesian model choice! (I had presented preliminary versions at the recent workshops in New York and Zürich.) While broader in scope, the results obtained by Judith Rousseau, Jean-Michel Marin, Natesh Pillai, and myself bring an answer to the question raised by our PNAS paper on ABC model choice. Almost as soon as we realised the problem, that is, during MCMC’Ski in Utah, I talked with Judith about a possible classification of statistics in terms of their Bayes factor performances and we started working on that… While the idea of separating the mean behaviour of the statistics under both model came rather early, establishing a complete theoretical framework that validated this intuition took quite a while and the assumptions changed a few times around the summer. The simulations associated with the paper were straightforward in that (a) the setup had been suggested to us by a referee of our PNAS paper: compare normal and Laplace distributions with different summary statistics (inc. the median absolute deviation), (b) the theoretical results told us what to look for, and (c) they did very clearly exhibit the consistency and inconsistency of the Bayes factor/posterior probability predicted by the theory. Both boxplots shown here exhibit this agreement: when using (empirical) mean, median, and variance to compare normal and Laplace models, the posterior probabilities do not select the “true” model but instead aggregate near a fixed value. When using instead the median absolute deviation as summary statistic, the posterior probabilities concentrate near one or zero depending on whether or not the normal model is the true model.
The main result states that, under some “heavy-duty” assumptions, (a) if the “true” mean of the summary statistic can be recovered for both models under comparison, then the Bayes factor has the same asymptotic behaviour as n to the power -(d1 – d2)/2, irrespective of which one is the true model. (The dimensions d1 and d2 are the effective dimensions of the asymptotic means of the summary statistic under both models.) Therefore, the Bayes factor always asymptotically selects the model having the smallest effective dimension and cannot be consistent. (b) if, instead, the “true” mean of the summary statistic cannot be represented in the other model, then the Bayes factor is consistent. This means that, somehow, the best statistics to be used in an ABC approximation to a Bayes factor are ancillary statistics with different mean values under both models. Else, the summary statistic must have enough components to prohibit a parameter under the “wrong” model to meet the “true” mean of the summary statistic.
(As a striking coincidence, Hélene Massam and Géard Letac [re]posted today on arXiv a paper about the behaviour of the Bayes factor for contingency tables when the hyperparameter goes to zero, where they establish the consistency of the said Bayes factor under the sparser model. No Jeffreys-Lindley paradox in that case.)
The colloquium in honour of our friend Hans Ruedi Künsch was superbly organised by Peter Bühlmann, Sara van de Geer, and Marloes Maathuis! It was a pleasure to hear the talks by the other speakers (although fog in Zürich made me miss half of Jim’s talk on volcanoes!), to ponder about Sue Geman’s conjecture on the number of local modes on a mixture likelihood surface, to attend a very pleasant gala dinner, and to celebrate Hans’ numerous contributions to the field and to the community. Not to mention a good recovery run along Zürichsee in the early morning… Here are the slides of my talk, with very little chances from those in Columbia last week (as I did not have time to include the verification of the assumptions in the Laplace example):
As posted in the recent entry about the colloquium for Mike Titterington’s retiral, I will attend another colloquium this year to mark Hans Künsch’s 60th birthday. Which will give me an opportunity to meet with friends in Zürich later this Autumn. And to celebrate Hans’ contributions to Statistics. Given our current research, I will talk about ABC extensions and validations.