Archive for deviance

A new approach to Bayesian hypothesis testing

Posted in Books, Statistics with tags , , , , , on September 8, 2016 by xi'an

“The main purpose of this paper is to develop a new Bayesian hypothesis testing approach for the point null hypothesis testing (…) based on the Bayesian deviance and constructed in a decision theoretical framework. It can be regarded as the Bayesian version of the likelihood ratio test.”

This paper got published in Journal of Econometrics two years ago but I only read it a few days ago when Kerrie Mengersen pointed it out to me. Here is an interesting criticism of Bayes factors.

“In the meantime, unfortunately, Bayes factors also suffers from several theoretical and practical difficulties. First, when improper prior distributions are used, Bayes factors contains undefined constants and takes arbitrary values (…) Second, when a proper but vague prior distribution with a large spread is used to represent prior ignorance, Bayes factors tends to favour the null hypothesis. The problem may persist even when the sample size is large (…) Third, the calculation of Bayes factors generally requires the evaluation of marginal likelihoods. In many models, the marginal likelihoods may be difficult to compute.”

I completely agree with these points, which are part of a longer list in our testing by mixture estimation paper. The authors also rightly blame the rigidity of the 0-1 loss function behind the derivation of the Bayes factor. An alternative decision-theoretic based on the Kullback-Leibler distance has been proposed by José Bernardo and Raúl Rueda, in a 2002 paper, evaluating the average divergence between the null and the full under the full, with the slight drawback that any nuisance parameter has the same prior under both hypotheses. (Which makes me think of the Savage-Dickey paradox, since everything here seems to take place under the alternative.) And the larger drawback of requiring a lower bound for rejecting the null. (Although it could be calibrated under the null prior predictive.)

This paper suggests using instead the difference of the Bayesian deviances, which is the expected log ratio integrated against the posterior. (With the possible embarrassment of the quantity having no prior expectation since the ratio depends on the data. But after all the evidence or marginal likelihood faces the same “criticism”.) So it is a sort of Bayes factor on the logarithms, with a strong similarity with Bernardo & Rueda’s solution since they are equal in expectation under the marginal. As in Dawid et al.’s recent paper, the logarithm removes the issue with the normalising constant and with the Lindley-Jeffreys paradox. The approach then needs to be calibrated in order to define a decision bound about the null. The asymptotic distribution of the criterion is  χ²(p)−p, where p is the dimension of the parameter to be tested, but this sounds like falling back on frequentist tests. And the deadly .05% bounds. I would rather favour a calibration of the criterion using prior or posterior predictives under both models…

reading classics (#1)

Posted in Books, Statistics, University life with tags , , , , , , , , , on October 31, 2013 by xi'an

Here we are, back in a new school year and with new students reading the classics. Today, Ilaria Masiani started the seminar with a presentation of Spiegelhalter et al. 2002 DIC paper, already heavily mentioned on this blog. Here are the slides, posted on slideshare (if you know of another website housing and displaying slides, let me know: the incompatibility between Firefox and slideshare drives me crazy!, well, almost…)

I have already said a lot about DIC on this blog so will not add a repetition of my reservations. I enjoyed the link with the log scores and the Kullback-Leibler divergence, but failed to see a proper intuition for defining the effective number of parameters the way it is defined in the paper… The presentation was quite faithful to the original and, as is usual in the reading seminars (esp. the first one of the year), did not go far enough (for my taste) in the critical evaluation of the themes therein. Maybe an idea for next year would be to ask one of my PhD students to give the zeroth seminar…

my DICussion

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , on September 25, 2013 by xi'an

IMG_1648Following the Re-Reading of Spiegelhalter et al. (2002) by David at the RSS Annual Conference a few weeks ago, and my invited discussion there, I was asked to contribute a written discussion to Series B, a request obviously impossible to refuse!

The main issue with DIC is the question of its worth for Bayesian decision analysis (since I doubt there are many proponents of DIC outside the Bayesian community). The appeal of DIC is, I presume, to deliver a single summary per model under comparison and to allow therefore for a complete ranking of those models. I however object at the worth of simplicity for simplicity’s sake: models are complex (albeit less than reality) and their usages are complex as well. To consider that model A is to be preferred upon model B just because DIC(A)=1228 < DiC(B)=1237 is a mimicry of the complex mechanisms at play behind model choice, especially given the wealth of information provided by a Bayesian framework. (Non-Bayesian paradigms are more familiar with procedures based on a single estimator value.) And to abstain from accounting for the significance of the difference between DIC(A) and DIC(B) clearly makes matters worse.

This is not even discussing the stylised setting where one model is considered as “true” and where procedures are compared by their ability to recover the “truth”. David Spiegelhalter repeatedly mentioned during his talk that he was not interested in this. This stance brings another objection, though, namely that models can only be compared against their predictive abilities, which DIC seems unable to capture. Once again, what is needed is a multi-factor and all-encompassing criterion that evaluates the predictive models in terms of their recovery of some features of the phenomenon under study. Or of the process being conducted. (Even stooping down to a one-dimensional loss function that is supposed to summarise the purpose of the model comparison does not produce anything close to the DIC function.)

ruins of the abbey at Tynemouth, Sept. 03, 2013Obviously, considering that asymptotic consistency is of no importance whatsoever (as repeated by David in Newcastle) avoids some embarrassing questions, except the one about the true purpose of statistical models and procedures. How can they be compared if no model is true and if accumulating data from a given model is not meaningful? How can simulation be conducted in such a barren landscape?  I find it the more difficult to accept this minimalist attitude that models are truly used as if they were or could be true, at several stages in the process. It also prevents the study of the criterion under model misspecification, which would clearly be of interest.

Another point, already exposed in our 2006 Bayesian Analysis paper with Gilles Celeux, Florence Forbes, and Mike Titterington, is that there is no unique driving principle for constructing DICs. In that paper, we examined eight different and natural versions of DIC for mixture models, resulting in highly diverging values for DIC and the effective dimension of the parameter, I believe that such a lack of focus is bound to reappear in any multimodal setting and fear that the answer about (eight) different focus on what matters in the model is too cursory and lacks direction for the hapless practitioner.

My final remark about DIC is that it shares very much the same perspective as Murray Aitkin’s integrated likelihood, Both Aitkin (1991, 2009) and Spiegelhalter et al. (2002) consider a posterior distribution on the likelihood function, taken as a function of the parameter but omitting the delicate fact that it also depends on the observable and hence does not exist a priori. We wrote a detailed review of Aitkin’s (2009) book, where most of the criticisms equally apply to DIC, and I will not repeat them here, except for pointing out that it escapes the Bayesian framework (and thus requires even more its own justifications).