Archive for improper posteriors

Mea Culpa

Posted in Statistics with tags , , , , , , , , , , , on April 10, 2020 by xi'an

[A quote from Jaynes about improper priors that I had missed in his book, Probability Theory.]

For many years, the present writer was caught in this error just as badly as anybody else, because Bayesian calculations with improper priors continued to give just the reasonable and clearly correct results that common sense demanded. So warnings about improper priors went unheeded; just that psychological phenomenon. Finally, it was the marginalization paradox that forced recognition that we had only been lucky in our choice of problems. If we wish to consider an improper prior, the only correct way of doing it is to approach it as a well-defined limit of a sequence of proper priors. If the correct limiting procedure should yield an improper posterior pdf for some parameter α, then probability theory is telling us that the prior information and data are too meager to permit any inferences about α. Then the only remedy is to seek more data or more prior information; probability theory does not guarantee in advance that it will lead us to a useful answer to every conceivable question.Generally, the posterior pdf is better behaved than the prior because of the extra information in the likelihood function, and the correct limiting procedure yields a useful posterior pdf that is analytically simpler than any from a proper prior. The most universally useful results of Bayesian analysis obtained in the past are of this type, because they tended to be rather simple problems, in which the data were indeed so much more informative than the prior information that an improper prior gave a reasonable approximation – good enough for all practical purposes – to the strictly correct results (the two results agreed typically to six or more significant figures).

In the future, however, we cannot expect this to continue because the field is turning to more complex problems in which the prior information is essential and the solution is found by computer. In these cases it would be quite wrong to think of passing to an improper prior. That would lead usually to computer crashes; and, even if a crash is avoided, the conclusions would still be, almost always, quantitatively wrong. But, since likelihood functions are bounded, the analytical solution with proper priors is always guaranteed to converge properly to finite results; therefore it is always possible to write a computer program in such a way (avoid underflow, etc.) that it cannot crash when given proper priors. So, even if the criticisms of improper priors on grounds of marginalization were unjustified,it remains true that in the future we shall be concerned necessarily with proper priors.

statistics with improper posteriors [or not]

Posted in Statistics with tags , , , , , , on March 6, 2019 by xi'an

Last December, Gunnar Taraldsen, Jarle Tufto, and Bo H. Lindqvist arXived a paper on using priors that lead to improper posteriors and [trying to] getting away with it! The central concept in their approach is Rényi’s generalisation of Kolmogorov’s version to define conditional probability distributions from infinite mass measures by conditioning on finite mass measurable sets. A position adopted by Dennis Lindley in his 1964 book .And already discussed in a few ‘Og’s posts. While the theory thus developed indeed allows for the manipulation of improper posteriors, I have difficulties with the inferential aspects of the construct, since one cannot condition on an arbitrary finite measurable set without prior information. Things get a wee bit more outwardly when considering “data” with infinite mass, in Section 4.2, since they cannot be properly normalised (although I find the example of the degenerate multivariate Gaussian distribution puzzling as it is not a matter of improperness, since the degenerate Gaussian has a well-defined density against the right dominating measure).  The paper also discusses marginalisation paradoxes, by acknowledging that marginalisation is no longer feasible with improper quantities. And the Jeffreys-Lindley paradox, with a resolution that uses the sum of the Dirac mass at the null, δ⁰, and of the Lebesgue measure on the real line, λ, as the dominating measure. This indeed solves the issue of the arbitrary constant in the Bayes factor, since it is “the same” on the null hypothesis and elsewhere, but I do not buy the argument, as I see no reason to favour δ⁰+λ over 3.141516 δ⁰+λ or δ⁰+1.61718 λ… (This section 4.5 also illustrates that the choice of the sequence of conditioning sets has an impact on the limiting measure, in the Rényi sense.) In conclusion, after reading the paper, I remain uncertain as to how to exploit this generalisation from an inferential (Bayesian?) viewpoint, since improper posteriors do not clearly lead to well-defined inferential procedures…

back to the Bayesian Choice

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , on October 17, 2018 by xi'an

Surprisingly (or not?!), I received two requests about some exercises from The Bayesian Choice, one from a group of students from McGill having difficulties solving the above, wondering about the properness of the posterior (but missing the integration of x), to whom I sent back this correction. And another one from the Czech Republic about a difficulty with the term “evaluation” by which I meant (pardon my French!) estimation.

improperties on an astronomical scale

Posted in Books, pictures, Statistics with tags , , , , , , , on December 15, 2017 by xi'an

As pointed out by Peter Coles on his blog, In the Dark, Hyungsuk Tak, Sujit Ghosh, and Justin Ellis just arXived a review of the unsafe use of improper priors in astronomy papers, 24 out of 75 having failed to establish that the corresponding posteriors are well-defined. And they exhibit such an instance (of impropriety) in a MNRAS paper by Pihajoki (2017), which is a complexification of Gelfand et al. (1990), also used by Jim Hobert in his thesis. (Even though the formal argument used to show the impropriety of the posterior in Pihajoki’s paper does not sound right since it considers divergence at a single value of a parameter β.) Besides repeating this warning about an issue that was rather quickly identified in the infancy of MCMC, if not in the very first publications on the Gibbs sampler, the paper seems to argue against using improper priors due to this potential danger, stating that instead proper priors that include all likely values and beyond are to be preferred. Which reminds me of the BUGS feature of using a N(0,10⁹) prior instead of the flat prior, missing the fact that “very large” variances do impact the resulting inference (if only for the issue of model comparison, remember Lindley-Jeffreys!). And are informative in that sense. However, it is obviously a good idea to advise checking for propriety (!) and using such alternatives may come as a safety button, providing a comparison benchmark to spot possible divergences in the resulting inference.

Jeffreys priors for mixtures [or not]

Posted in Books, Statistics, University life with tags , , , , , on July 25, 2017 by xi'an

Clara Grazian and I have just arXived [and submitted] a paper on the properties of Jeffreys priors for mixtures of distributions. (An earlier version had not been deemed of sufficient interest by Bayesian Analysis.) In this paper, we consider the formal Jeffreys prior for a mixture of Gaussian distributions and examine whether or not it leads to a proper posterior with a sufficient number of observations.  In general, it does not and hence cannot be used as a reference prior. While this is a negative result (and this is why Bayesian Analysis did not deem it of sufficient importance), I find it definitely relevant because it shows that the default reference prior [in the sense that the Jeffreys prior is the primary choice in nonparametric settings] does not operate in this wide class of distributions. What is surprising is that the use of a Jeffreys-like prior on a global location-scale parameter (as in our 1996 paper with Kerrie Mengersen or our recent work with Kaniav Kamary and Kate Lee) remains legit if proper priors are used on all the other parameters. (This may be yet another illustration of the tequilla-like toxicity of mixtures!)

Francisco Rubio and Mark Steel already exhibited this difficulty of the Jeffreys prior for mixtures of densities with disjoint supports [which reveals the mixture latent variable and hence turns the problem into something different]. Which relates to another point of interest in the paper, derived from a 1988 [Valencià Conference!] paper by José Bernardo and Javier Giròn, where they show the posterior associated with a Jeffreys prior on a mixture is proper when (a) only estimating the weights p and (b) using densities with disjoint supports. José and Javier use in this paper an astounding argument that I had not seen before and which took me a while to ingest and accept. Namely, the Jeffreys prior on a observed model with latent variables is bounded from above by the Jeffreys prior on the corresponding completed model. Hence if the later leads to a proper posterior for the observed data, so does the former. Very smooth, indeed!!!

Actually, we still support the use of the Jeffreys prior but only for the mixture mixtures, because it has the property supported by Judith and Kerrie of a conservative prior about the number of components. Obviously, we cannot advocate its use over all the parameters of the mixture since it then leads to an improper posterior.