While I could not find an on-line picture of Mike Titterington, another testimony to his modesty and selflessness, I remembered this series of sketches on priors he made for the Bulletin in Applied Statistics in 1982, under the title Irreverent Bayes!
Another paper on Lindley’s paradox appeared on arXiv yesterday, by Guosheng Yin and Haolun Shi, interpreting posterior probabilities as p-values. The core of this resolution is to express a two-sided hypothesis as a combination of two one-sided hypotheses along the opposite direction, taking then advantage of the near equivalence of posterior probabilities under some non-informative prior and p-values in the later case. As already noted by George Casella and Roger Berger (1987) and presumably earlier. The point is that one-sided hypotheses are quite friendly to improper priors, since they only require a single prior distribution. Rather than two when point nulls are under consideration. The p-value created by merging both one-sided hypotheses makes little sense to me as it means testing that both θ≥0 and θ≤0, resulting in the proposal of a p-value that is twice the minimum of the one-sided p-values, maybe due to a Bonferroni correction, although the true value should be zero… I thus see little support for this approach to resolving Lindley paradox in that it bypasses the toxic nature of point-null hypotheses that require a change of prior toward a mixture supporting one hypothesis and the other. Here the posterior of the point-null hypothesis is defined in exactly the same way the p-value is defined, hence making the outcome most favourable to the agreement but not truly addressing the issue.
Our paper, weakly informative reparameterisations of location-scale mixtures, with Kaniav Kamary and Kate Lee, got accepted by JCGS! Great news, which comes in perfect timing for Kaniav as she is currently applying for positions. The paper proposes a unidimensional mixture Bayesian modelling based on the first and second moment constraints, since these turn the remainder of the parameter space into a compact. While we had already developed an associated R package, Ultimixt, the current editorial policy of JCGS imposes the R code used to produce all results to be attached to the submission and it took us a few more weeks than it should have to produce a directly executable code, due to internal library incompatibilities. (For this entry, I was looking for a link to our special JCGS issue with my picture of Edinburgh but realised I did not have this picture.)
Dexter Cahoy and Joseph Sedransk just arXived a paper on so-called inverse stable priors. The starting point is the supposed defficiency of Gamma conjugate priors, which have explosive behaviour near zero. Albeit remaining proper. (This behaviour eventually vanishes for a large enough sample size.) The alternative involves a transform of alpha-stable random variables, with the consequence that the density of this alternative prior does not have a closed form. Neither does the posterior. When the likelihood can be written as exp(a.θ+b.log θ), modulo a reparameterisation, which covers a wide range of distributions, the posterior can be written in terms of the inverse stable density and of another (intractable) function called the generalized Mittag-Leffler function. (Which connects this post to an earlier post on Sofia Kovaleskaya.) For simulating this posterior, the authors suggest using an accept-reject algorithm based on the prior as proposal, which has the advantage of removing the intractable inverse stable density but the disadvantage of… simulating from the prior! (No mention is made of the acceptance rate.) I am thus reserved as to how appealing this new proposal is, despite “the inverse stable density (…) becoming increasingly popular in several areas of study”. And hence do not foresee a bright future for this class of prior…
E.J. Wagenmakers and his enthusiastic team of collaborators at University of Amsterdam and in the JASP software designing team have started a blog called Bayesian spectacles which I find a fantastic title. And not only because I wear glasses. Plus, they got their own illustrator, Viktor Beekman, which sounds like the epitome of sophistication! (Compared with resorting to vacation or cat pictures…)
In a most recent post they addressed the criticisms we made of the 72 author paper on p-values, one of the co-authors being E.J.! Andrew already re-addressed some of the address, but here is a disagreement he let me to chew on my own [and where the Abandoners are us!]:
Disagreement 2. The Abandoners’ critique the UMPBTs –the uniformly most powerful Bayesian tests– that features in the original paper. This is their right (see also the discussion of the 2013 Valen Johnson PNAS paper), but they ignore the fact that the original paper presented a series of other procedures that all point to the same conclusion: p-just-below-.05 results are evidentially weak. For instance, a cartoon on the JASP blog explains the Vovk-Sellke bound. A similar result is obtained using the upper bounds discussed in Berger & Sellke (1987) and Edwards, Lindman, & Savage (1963). We suspect that the Abandoners’ dislike of Bayes factors (and perhaps their upper bounds) is driven by a disdain for the point-null hypothesis. That is understandable, but the two critiques should not be mixed up. The first question is Given that we wish to test a point-null hypothesis, do the Bayes factor upper bounds demonstrate that the evidence is weak for p-just-below-.05 results? We believe they do, and in this series of blog posts we have provided concrete demonstrations.
Obviously, this reply calls for an examination of the entire BS blog series, but being short in time at the moment, let me point out that the upper lower bounds on the Bayes factors showing much more support for H⁰ than a p-value at 0.05 only occur in special circumstances. Even though I spend some time in my book discussing those bounds. Indeed, the [interesting] fact that the lower bounds are larger than the p-values does not hold in full generality. Moving to a two-dimensional normal with potentially zero mean is enough to see the order between lower bound and p-value reverse, as I found [quite] a while ago when trying to expand Berger and Sellker (1987, the same year as I was visiting Purdue where both had a position). I am not sure this feature has been much explored in the literature, I did not pursue it when I realised the gap was missing in larger dimensions… I must also point out I do not have the same repulsion for point nulls as Andrew! While considering whether a parameter, say a mean, is exactly zero [or three or whatever] sounds rather absurd when faced with the strata of uncertainty about models, data, procedures, &tc.—even in theoretical physics!—, comparing several [and all wrong!] models with or without some parameters for later use still makes sense. And my reluctance in using Bayes factors does not stem from an opposition to comparing models or from the procedure itself, which is quite appealing within a Bayesian framework [thus appealing per se!], but rather from the unfortunate impact of the prior [and its tail behaviour] on the quantity and on the delicate calibration of the thing. And on a lack of reference solution [to avoid the O and the N words!]. As exposed in the demise papers. (Which main version remains in a publishing limbo, the onslaught from the referees proving just too much for me!)
Xi'an's Og 