Bayes’s theorem for improper mixtures

While looking for references for a Master summer project at Warwick on Bayesian inference on the Cauchy location parameter, I came across a 2011 Annals of Statistics paper by Peter McCullagh and Han Han.  Which expands the Bayesian framework to the improper case by considering a Poisson process over the parameter set with mean measure ν the improper prior. Instead of a single random parameter, this construct returns a countable collection of pairs (θ,y), while the observations induce a subset of that collection constrained by y∈A, a “sampling region” both capital to the derivation of the joint distribution and obscure in that A remains unspecified (but such that 0<ν(A)<∞ and conveniently returning the observed sample of y’s).

“Provided that the key finiteness condition is satisfied, this probabilistic analysis of the extended model may be interpreted as a vindication of improper Bayes procedures derived from the original model.”

“Thus, the existence of a joint probability model associated with an improper prior does not imply optimality in the form of coherence, consistency or admissibility.”

This is definitely fascinating!, even though I have troubles linking this infinite sequence of θ‘s with regular Bayesian inference, since the examples in the paper seem to revert to a single parameter value, as in §4.1, for the Normal model and §5 for the Cauchy model. The authors also revisit the marginalisation paradoxes of Dawid, Stone and Zidek (1973), with the argument that the improper measure leading to the paradox is not compatible with ν(A)<∞, hence does not define a natural conditional, while the “other” improper measure avoids the paradox.

prior sensitivity of the marginal likelihood

Fernando Llorente and (Madrilene) coauthors have just arXived a paper on the safe use of prior densities for Bayesian model selection. Rather than blaming the Bayes factor, or excommunicating some improper priors, they consider in this survey solutions to design “objective” priors in model selection. (Writing this post made me realised I had forgotten to arXive a recent piece I wrote on the topic, based on short courses and blog pieces, for an incoming handbook on Bayesian advance(ment)s! Soon to be corrected.)

While intrinsically interested in the topic and hence with the study, I somewhat disagree with the perspective adopted by the authors. They for instance stick to the notion that a flat prior over the parameter space is appropriate as “the maximal expression of a non-informative prior” (despite depending on the parameterisation). Over bounded sets at least, while advocating priors “with great scale parameter” otherwise. They also refer to Jeffreys (1939) priors, by which they mean estimation priors rather than testing priors. As uncovered by Susie Bayarri and Gonzalo Garcia-Donato. Considering asymptotic consistency, they state that “in the asymptotic regime, Bayesian model selection is more sensitive to the sample size D than to the prior specifications”, which I find both imprecise and confusing,  as my feeling is that the prior specification remains overly influential as the sample size increases. (In my view, consistency is a minimalist requirement, rather than “comforting”.) The argument therein that a flat prior is informative for model choice stems from the fact that the marginal likelihood goes to zero as the support of the prior goes to infinity, which may have been an earlier argument of Jeffreys’ (1939), but does not carry much weight as the property is shared by many other priors (as remarked later). Somehow, the penalisation aspect of the marginal is not exploited more deeply in the paper. In the “objective” Bayes section, they adhere to the (convenient but weakly supported) choice of a common prior on the nuisance parameters (shared by different models). Their main argument is to develop (heretic!) “data-based priors”, from Aitkin (1991, not cited) double use of the data (or setting the likelihood to the power two), all the way to the intrinsic and fractional Bayes factors of Tony O’Hagan (1995), Jim Berger and Luis Pericchi (1996), and to the expected posterior priors of Pérez and Berger (2002) on which I worked with Juan Cano and Diego Salmeròn. (While the presentation is made against a flat prior, nothing prevents the use of another reference, improper, prior.) A short section also mentions the X-validation approach(es) of Aki Vehtari and co-authors.

nested sampling: any prior anytime?!

A recent arXival by Justin Alsing and Will Handley on “nested sampling with any prior you like” caught my attention. If only because I was under the impression that some priors would not agree with nested sampling. Especially those putting positive weight on some fixed levels of the likelihood function, as well as improper priors.

“…nested sampling has largely only been practical for a somewhat restrictive class of priors, which have a readily available representation as a transform from the unit hyper-cube.”

Reading from the paper, it seems that the whole point is to demonstrate that “any proper prior may be transformed onto the unit hypercube via a bijective transformation.” Which seems rather straightforward if the transform is not otherwise constrained: use a logit transform in every direction. The paper gets instead into the rather fashionable direction of normalising flows as density representations. (Which suddenly reminded me of the PhD dissertation of Rob Cornish at Oxford, which I examined last year. Even though nested was not used there in the same understanding.) The purpose appearing later (in the paper) or in fine to express a random variable simulated from the prior as the (generative) transform of a Uniform variate, f(U). Resuscitating the simulation from an arbitrary distribution from first principles.

“One particularly common scenario where this arises is when one wants to use the (sampled) posterior from one experiment as the prior for another”

But I remained uncertain at the requirement for this representation in implementing nested sampling as I do not see how it helps in bypassing the hurdles of simulating from the prior constrained by increasing levels of the likelihood function. It would be helpful to construct normalising flows adapted to the truncated priors but I did not see anything related to this version in the paper.

The cosmological application therein deals with the incorporation of recent measurements in the study of the ΛCDM cosmological model, that is, more recent that the CMB Planck dataset we played with 15 years ago. (Time flies, even if an expanding Universe!) Namely, the Baryon Oscillation Spectroscopic Survey and the SH₀ES collaboration.

logic (not logistic!) regression

A Bayesian Analysis paper by Aliaksandr Hubin, Geir Storvik, and Florian Frommlet on Bayesian logic regression was open for discussion. Here are some hasty notes I made during our group discussion in Paris Dauphine (and later turned into a discussion submitted to Bayesian Analysis):

“Originally logic regression was introduced together with likelihood based model selection, where simulated annealing served as a strategy to obtain one “best” model.”

Indeed, logic regression is not to be confused with logistic regression! Rejection of a true model in Bayesian model choice leads to Bayesian model choice and… apparently to Bayesian logic regression. The central object of interest is a generalised linear model based on a vector of binary covariates and using some if not all possible logical combinations (trees) of said covariates (leaves). The GLM is further using rather standard indicators to signify whether or not some trees are included in the regression (and hence the model). The prior modelling on the model indices sounds rather simple (simplistic?!) in that it is only function of the number of active trees, leading to an automated penalisation of larger trees and not accounting for a possible specificity of some covariates. For instance when dealing with imbalanced covariates (much more 1 than 0, say).

A first question is thus how much of a novel model this is when compared with say an analysis of variance since all covariates are dummy variables. Culling the number of trees away from the exponential of exponential number of possible covariates remains obscure but, without it, the model is nothing but variable selection in GLMs, except for “enjoying” a massive number of variables. Note that there could be a connection with variable length Markov chain models but it is not exploited there.

“…using Jeffrey’s prior for model selection has been widely criticized for not being consistent once the true model coincides with the null model.”

A second point that strongly puzzles me in the paper is its loose handling of improper priors. It is well-known that improper priors are at worst fishy in model choice settings and at best avoided altogether, to wit the Lindley-Jeffreys paradox and friends. Not only does the paper adopts the notion of a same, improper, prior on the GLM scale parameter, which is a position adopted in some of the Bayesian literature, but it also seems to be using an improper prior on each set of parameters (further undifferentiated between models). Because the priors operate on different (sub)sets of parameters, I think this jeopardises the later discourse on the posterior probabilities of the different models since they are not meaningful from a probabilistic viewpoint, with no joint distribution as a reference, neither marginal density. In some cases, p(y|M) may become infinite. Referring to a “simple Jeffrey’s” prior in this setting is therefore anything but simple as Jeffreys (1939) himself shied away from using improper priors on the parameter of interest. I find it surprising that this fundamental and well-known difficulty with improper priors in hypothesis testing is not even alluded to in the paper. Its core setting thus seems to be flawed. Now, the numerical comparison between Jeffrey’s [sic] prior and a regular g-prior exhibits close proximity and I thus wonder at the reason. Could it be that the culling and selection processes end up having the same number of variables and thus eliminate the impact of the prior? Or is it due to the recourse to a Laplace approximation of the marginal likelihood that completely escapes the lack of definition of the said marginal? Computing the normalising constant and repeating this computation while the algorithm is running ignores the central issue.

“…hereby, all states, including all possible models of maximum sized, will eventually be visited.”

Further, I found some confusion between principles and numerics. And as usual bemoan the acronym inflation with the appearance of a GMJMCMC! Where G stands for genetic (algorithm), MJ for mode jumping, and MCMC for…, well no surprise there! I was not aware of the mode jumping algorithm of Hubin and Storvik (2018), so cannot comment on the very starting point of the paper. A fundamental issue with Markov chains on discrete spaces is that the notion of neighbourhood becomes quite fishy and is highly dependent on the nature of the covariates. And the Markovian aspects are unclear because of the self-avoiding aspect of the algorithm. The novel algorithm is intricate and as such seems to require a superlative amount of calibration. Are all modes truly visited, really? (What are memetic algorithms?!)

O’Bayes 19/4

Last talks of the conference! With Rui Paulo (along with Gonzalo Garcia-Donato) considering the special case of factors when doing variable selection. Which is an interesting question that I had never considered, as at best I would remove all leves or keeping them all. Except that there may be misspecification in the factors as for instance when several levels have the same impact.With Michael Evans discussing a paper that he wrote for the conference! Following his own approach to statistical evidence. And including his reluctance to cover infinity (calling on Gauß for backup!) or continuity, and his call to falsify a Bayesian model by checking it can be contradicted by the data. His assumption that checking for prior is separable from checking for [sampling] model is debatable. (With another mention made of the Savage-Dickey ratio.)

And with Dimitris Fouskakis giving a wide ranging assessment [which Mark Steel (Warwick) called a PEP talk!] of power-expected-posterior priors, used with reference (and usually improper) priors. Which in retrospect would have suited better the beginning of the conference as it provided a background to several of the talks. Raising a question (from my perspective) on using the maximum likelihood estimator as a pseudo-sufficient statistic when this MLE is computed for the base (simplest) model. Maybe an ABC induced bias in this question as it would not work for ABC model choice.

Overall, I think the scientific outcomes of the conference were quite positive: a wide range of topics and perspectives, a reasonable and diverse attendance, especially when considering the heavy load of related conferences in the surrounding weeks (the “June fatigue”!), animated poster sessions. I am obviously not the one to assess the organisation of the conference! Things I forgot to do in this regard: organise transportation from Oxford to Warwick University, provide an attached room for in-pair research, insist on sustainability despite the imposed catering solution, facilitate sharing joint transportation to and from the Warwick campus, mention that tap water was potable, and… wear long pants when running in nettles.