Archive for improper prior

« Older Entries

a lesser-known correlate of the Jeffreys-Lindley paradox (with discussion)

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , on October 19, 2024 by xi'an

Two UBC faculty, Harlan Campbell and Paul Gustafson, wrote a paper entitled “Defining a Credible Interval Is Not Always Possible with “Point-Null” Priors: A Lesser-Known Correlate of the Jeffreys-Lindley Paradox” in Bayesian Analysis (2024, 19, Number 3, pp. 925–984), which got discussed and presented on the BA webinar yesterday. I missed the call for discussion, on a topic I would have liked very much to discuss and an analysis I strongly disagree with. Fortunately, several of the discussants in the webinar and in the printed version advanced some of my points (as. e.g., Bertrand Clarke in the above slide screen-shot from the on-line video).

I find the paper somewhat missing in linking with the history of the topic, with no mention of Berger & Sellke (1987) that comes as a counterpoint to Casella &—the other—Berger (1987), opposing one sided to two sided tests. Or of matching priors, which connect credible and confidence intervals to higher orders. But the central issue with the apparent contradiction between rejecting the point null hypothesis and returning a credible interval that contains the null is that the construction proceeds from a model averaged posterior. Which fundamentally contradicts the construct of a pair of priors attached with each model towards selecting the fittest one. And requires a far-from-innocent choice of respective prior weights for both models, an ill-defined notion I have repeatedly criticised here and elsewhere. Model averaging clashes with model selection in both decision-theoretic and modelling terms. In model averaging terms, the disappearance of the opposition exhibited by the authors in the predictive distribution, as shown by discussants Held and Pawel, is unsurprising. And makes the spike-and-slab prior far of a necessity. Contrariwise to the model selection case where it proves unavoidable. And for which a merged credible interval does not make sense (to me at least) since it should be constructed once one (and only one) of the two models is chosen. At this point, that the other model ever was considered should not impact subsequent inference. And within that perspective I do not see the relevance of agnostic (ignoring the model choice ation) 5% confidence or credible regions.

“…considers the regime of a fixed true parameter value as n increases [and] of a fixed p-value…” (p928)

With regards with the connection with the Jeffreys-Lindley (or Lindley-Jeffreys) so-called paradox, on which I have already written a lot (or even too much!), many of the earlier objections resurface. Like the measure-theoretic difficulty in including within a continuous interval an atom, i.e., a value with a point mass. Which isolates this atom away from any other value in the interval (and of course creates discontinuities). Or fixing the p-value forever after (when n goes to infinity), as in the graph below (p929). Or treating an improper prior without further caution than with a proper prior. Especially when these are “created” by the decision problem itself.

 

2 Comments »

Bayes Factors for Forensic Decision Analyses with R [book review]

Posted in Books, R, Statistics with tags , , , , , , , , , , , , , on November 28, 2022 by xi'an

My friend EJ Wagenmaker pointed me towards an entire book on the BF by Bozza (from Ca’Foscari, Venezia), Taroni and Biederman. It is providing a sort of blueprint for using Bayes factors in forensics for both investigative and evaluative purposes. With R code and free access. I am of course unable to judge of the relevance of the approach for forensic science (I was under the impression that Bayesian arguments were usually not well-received in the courtroom) but find that overall the approach is rather one of repositioning the standard Bayesian tools within a forensic framework.

“The [evaluative] purpose is to assign a value to the result of a comparison between an item of unknown source and an item from a known source.”

And thus I found nothing shocking or striking from this standard presentation of Bayes factors, including the call to loss functions, if a bit overly expansive in its exposition. The style is also classical, with a choice of grey background vignettes for R coding parts that we also picked in our R books! If anything, I would have expected more realistic discussions and illustrations of prior specification across the hypotheses (see e.g. page 34), while the authors are mostly centering on conjugate priors and the (de Finetti) trick of the equivalent prior sample size. Bayes factors are mostly assessed using a conservative version of Jeffreys’ “scale of evidence”. The computational section of the book introduces MCMC (briefly) and mentions importance sampling, harmonic mean (with a minimalist warning), and Chib’s formula (with no warning whatsoever).

“The [investigative] purpose is to provide information in investigative proceedings (…) The scientist (…) uses the findings to generate hypotheses and suggestions for explanations of observations, in order to give guidance to investigators or litigants.”

Chapter 2 is about standard models: inferring about a proportion, with some Monte Carlo illustration,  and the complication of background elements, normal mean, with an improper prior making an appearance [on p.69] with no mention being made of the general prohibition of such generalised priors when using Bayes factors or even of the Lindley-Jeffreys paradox. Again, the main difference with Bayesian textbooks stands with the chosen examples.

Chapter 3 focus on evidence evaluation [not in the computational sense] but, again, the coverage is about standard models: processing the Binomial, multinomial, Poisson models, again though conjugates. (With the side remark that Fig 3.2 is rather unhelpful: when moving the prior probability of the null from zero to one, its posterior probability also moves from zero to one!) We are back to the Normal mean case with the model variance being known then unknown. (An unintentionally funny remark (p.96) about the dependence between mean and variance being seen as too restrictive and replaced with… independence!). At last (for me!), the book is pointing [p.99] out that the BF is highly sensitive to the choice of the prior variance (Lindley-Jeffreys, where art thou?!), but with a return of the improper prior (on said variance, p.102) with no debate on the ensuing validity of the BF. Multivariate Normals are also presented, with Wishart priors on the precision matrix, and more details about Chib’s estimate of the evidence. This chapter also contains illustrations of the so-called score-based BF which is simply (?) a Bayes factor using a distribution on a distance summary (between an hypothetical population and the data) and an approximation of the distributions of these summaries, provided enough data is available… I also spotted a potentially interesting foray into BF variability (Section 3.4.2), although not reaching all the way to a notion of BF posterior distributions.

Chapter 4 stands for Bayes factors for investigation, where alternative(s) is(are) less specified, as testing eg Basmati rice vs non-Basmati rice. But there is no non-parametric alternative considered in the book. Otherwise, it looks to me rather similar to Chapter 3, i.e. being back to binomial, multinomial models, with more discussions onm prior specification, more normal, or non-normal model, where the prior distribution is puzzingly estimated by a kernel density estimator, a portmanteau alternative (p.157), more multivariate Normals with Wishart priors and an entry on classification & discrimination.

[Disclaimer about potential self-plagiarism: this post or an edited version will eventually appear in my Books Review section in CHANCE. As appropriate for a book about Chance!]

3 Comments »

[more than] everything you always wanted to know about marginal likelihood

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , , , , , , , , , on February 10, 2022 by xi'an

Earlier this year, F. Llorente, L. Martino, D. Delgado, and J. Lopez-Santiago have arXived an updated version of their massive survey on marginal likelihood computation. Which I can only warmly recommend to anyone interested in the matter! Or looking for a base camp to initiate a graduate project. They break the methods into four families

  1. Deterministic approximations (e.g., Laplace approximations)
  2. Methods based on density estimation (e.g., Chib’s method, aka the candidate’s formula)
  3. Importance sampling, including sequential Monte Carlo, with a subsection connecting with MCMC
  4. Vertical representations (mostly, nested sampling)

Besides sheer computation, the survey also broaches upon issues like improper priors and alternatives to Bayes factors. The parts I would have done in more details are reversible jump MCMC and the long-lasting impact of Geyer’s reverse logistic regression (with the noise contrasting extension), even though the link with bridge sampling is briefly mentioned there. There is even a table reporting on the coverage of earlier surveys. Of course, the following postnote of the manuscript

The Christian Robert’s blog deserves a special mention , since Professor C. Robert has devoted several entries of his blog with very interesting comments regarding the marginal likelihood estimation and related topics.

does not in the least make me less objective! Some of the final recommendations

  • use of Naive Monte Carlo [simulate from the prior] should be always considered [assuming a proper prior!]
  • a multiple-try method is a good choice within the MCMC schemes
  • optimal umbrella sampling estimator is difficult and costly to implement , so its best performance may not be achieved in practice
  • adaptive importance sampling uses the posterior samples to build a suitable normalized proposal, so it benefits from localizing samples in regions of high posterior probability while preserving the properties of standard importance sampling
  • Chib’s method is a good alternative, that provide very good performances [but is not always available]
  • the success [of nested sampling] in the literature is surprising.

1 Comment »

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.

3 Comments »

demystify Lindley’s paradox [or not]

Posted in Statistics with tags , , , , , on March 18, 2020 by xi'an

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.

Leave a comment »

« Older Entries