**D**exter 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…

## Archive for reference priors

## inverse stable priors

Posted in Statistics with tags All the pretty horses, alpha-stable processes, conjugate priors, Gösta Mittag-Leffler, non-informative priors, reference priors, Sofia Kovalevskaya on November 24, 2017 by xi'an## Bayesian spectacles

Posted in Books, pictures, Statistics, University life with tags Amsterdam, Bayes factors, Bayesian Spectacles, blogging, Holland, JASP, non-informative priors, objective Bayes, reference priors, UMPBTs, uniformly most powerful tests, University of Amsterdam on October 4, 2017 by xi'anE.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!)

## priors without likelihoods are like sloths without…

Posted in Books, Statistics with tags Austin, Bayes factors, Bayesian Analysis, identifiability, improper priors, noninformative priors, O'Bayes17, Pierre Simon Laplace, posterior predictive, reference priors, sloth, The American Statistician, The University of Texas at Austin on September 11, 2017 by xi'an

“The idea of building priors that generate reasonable data may seem like an unusual idea…”

**A**ndrew, Dan, and Michael arXived a opinion piece last week entitled “The prior can generally only be understood in the context of the likelihood”. Which connects to the earlier Read Paper of Gelman and Hennig I discussed last year. I cannot state strong disagreement with the positions taken in this piece, actually, in that I do not think prior distributions ever occur as *a given* but are rather chosen as a reference measure to probabilise the parameter space and eventually prioritise regions over others. If anything I find myself even further on the prior agnosticism gradation. (Of course, this lack of disagreement applies to the likelihood understood as a function of both the data and the parameter, rather than of the parameter only, conditional on the data. Priors cannot be depending on the data without incurring disastrous consequences!)

“…it contradicts the conceptual principle that the prior distribution should convey only information that is available before the data have been collected.”

The first example is somewhat disappointing in that it revolves as so many Bayesian textbooks (since Laplace!) around the [sex ratio] Binomial probability parameter and concludes at the strong or long-lasting impact of the Uniform prior. I do not see much of a contradiction between the use of a Uniform prior and the collection of prior information, if only because there is not standardised way to transfer prior information into prior construction. And more fundamentally because a parameter rarely makes sense by itself, alone, without a model that relates it to potential data. As for instance in a regression model. More, following my epiphany of last semester, about the relativity of the prior, I see no damage in the prior being relevant, as I only attach a *relative* meaning to statements based on the posterior. Rather than trying to limit the impact of a prior, we should rather build assessment tools to measure this impact, for instance by prior predictive simulations. And this is where I come to quite agree with the authors.

“…non-identifiabilities, and near nonidentifiabilites, of complex models can lead to unexpected amounts of weight being given to certain aspects of the prior.”

Another rather straightforward remark is that non-identifiable models see the impact of a prior remain as the sample size grows. And I still see no issue with this fact in a relative approach. When the authors mention (p.7) that purely mathematical priors perform more poorly than weakly informative priors it is hard to see what they mean by this “performance”.

“…judge a prior by examining the data generating processes it favors and disfavors.”

Besides those points, I completely agree with them about the fundamental relevance of the prior as a generative process, only when the likelihood becomes available. And simulatable. (This point is found in many references, including our response to the American Statistician paper *Hidden dangers of specifying noninformative priors*, with Kaniav Kamary. With the same illustration on a logistic regression.) I also agree to their criticism of the marginal likelihood and Bayes factors as being so strongly impacted by the choice of a prior, if treated as absolute quantities. I also if more reluctantly and somewhat heretically see a point in using the posterior predictive for assessing whether a prior is relevant for the data at hand. At least at a conceptual level. I am however less certain about how to handle improper priors based on their recommendations. In conclusion, it would be great to see one [or more] of the authors at O-Bayes 2017 in Austin as I am sure it would stem nice discussions there! (And by the way I have no prior idea on how to conclude the comparison in the title!)

## Jeffreys priors for mixtures [or not]

Posted in Books, Statistics, University life with tags Bayesian Analysis, improper posteriors, Jeffreys priors, mixtures of distributions, noninformative priors, reference priors on July 25, 2017 by xi'an**C**lara 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.

## same data – different models – different answers

Posted in Books, Kids, Statistics, University life with tags Beaumont-en-Auge, Jeffreys priors, Laplace succession rule, non-informative priors, Normandie, Pierre Simon de Laplace, reference priors, statue, The Bayesian Choice, wikipedia on June 1, 2016 by xi'an**A**n interesting question from a reader of the Bayesian Choice came out on X validated last week. It was about Laplace’s succession rule, which I found somewhat over-used, but it was nonetheless interesting because the question was about the discrepancy of the “non-informative” answers derived from two models applied to the data: an Hypergeometric distribution in the Bayesian Choice and a Binomial on Wikipedia. The originator of the question had trouble with the difference between those two “non-informative” answers as she or he believed that there was a *single* non-informative principle that should lead to a unique answer. This does not hold, even when following a reference prior principle like Jeffreys’ invariant rule or Jaynes’ maximum entropy tenets. For instance, the Jeffreys priors associated with a Binomial and a Negative Binomial distributions differ. And even less when considering that there is no unity in reaching those reference priors. (Not even mentioning the issue of the reference dominating measure for the definition of the entropy.) This led to an informative debate, which is the point of X validated.

On a completely unrelated topic, the survey ship looking for the black boxes of the crashed EgyptAir plane is called the Laplace.

## covariant priors, Jeffreys and paradoxes

Posted in Books, Statistics, University life with tags evidence, Harold Jeffreys, hierarchical Bayesian modelling, improper priors, inadmissibility, invariance, Jeffreys priors, marginalisation paradoxes, Neyman-Scott problem, noninformative priors, over-interpretation of improper priors, reference priors on February 9, 2016 by xi'an

“If no information is available, π(α|M) must not deliver information about α.”

**I**n a recent arXival apparently submitted to Bayesian Analysis, Giovanni Mana and Carlo Palmisano discuss of the choice of priors in metrology. Which reminded me of this meeting I attended at the Bureau des Poids et Mesures in Sèvres where similar debates took place, albeit being led by ferocious anti-Bayesians! Their reference prior appears to be the Jeffreys prior, because of its reparameterisation invariance.

“The relevance of the Jeffreys rule in metrology and in expressing uncertainties in measurements resides in the metric invariance.”

This, along with a second order approximation to the Kullback-Leibler divergence, is indeed one reason for advocating the use of a Jeffreys prior. I at first found it surprising that the (usually improper) prior is used in a marginal likelihood, as it cannot be normalised. A source of much debate [and of our alternative proposal].

“To make a meaningful posterior distribution and uncertainty assessment, the prior density must be covariant; that is, the prior distributions of different parameterizations must be obtained by transformations of variables. Furthermore, it is necessary that the prior densities are proper.”

The above quote is quite interesting both in that the notion of *covariant* is used rather than *invariant* or *equivariant*. And in that properness is indicated as a requirement. (Even more surprising is the noun associated with covariant, since it clashes with the usual notion of covariance!) They conclude that the marginal associated with an improper prior is null because the normalising constant of the prior is infinite.

“…the posterior probability of a selected model must not be null; therefore, improper priors are not allowed.”

Maybe not so surprisingly given this stance on improper priors, the authors cover a collection of “paradoxes” in their final and longest section: most of which makes little sense to me. First, they point out that the reference priors of Berger, Bernardo and Sun (2015) are not invariant, but this should not come as a surprise given that they focus on parameters of interest versus nuisance parameters. The second issue pointed out by the authors is that under Jeffreys’ prior, the posterior distribution of a given normal mean for n observations is a *t* with n degrees of freedom while it is a *t* with n-1 degrees of freedom from a frequentist perspective. This is not such a paradox since both distributions work in different spaces. Further, unless I am confused, this is one of the marginalisation paradoxes, which more straightforward explanation is that marginalisation is not meaningful for improper priors. A third paradox relates to a contingency table with a large number of cells, in that the posterior mean of a cell probability goes as the number of cells goes to infinity. (In this case, Jeffreys’ prior is proper.) Again not much of a bummer, there is simply not enough information in the data when faced with a infinite number of parameters. Paradox #4 is the Stein paradox, when estimating the squared norm of a normal mean. Jeffreys’ prior then leads to a constant bias that increases with the dimension of the vector. Definitely a bad point for Jeffreys’ prior, except that there is no Bayes estimator in such a case, the Bayes risk being infinite. Using a renormalised loss function solves the issue, rather than introducing as in the paper uniform priors on intervals, which require hyperpriors without being particularly compelling. The fifth paradox is the Neyman-Scott problem, with again the Jeffreys prior the culprit since the estimator of the variance is inconsistent. By a multiplicative factor of 2. Another stone in Jeffreys’ garden [of forking paths!]. The authors consider that the prior gives zero weight to any interval not containing zero, as if it was a proper probability distribution. And “solve” the problem by avoid zero altogether, which requires of course to specify a lower bound on the variance. And then introducing another (improper) Jeffreys prior on that bound… The last and final paradox mentioned in this paper is one of the marginalisation paradoxes, with a bizarre explanation that since the mean and variance μ and σ are not independent a posteriori, “the information delivered by x̄ should not be neglected”.

## mixtures are slices of an orange

Posted in Kids, R, Statistics with tags CFE 2015, Gaussian mixture, hyperparameter, improper priors, invariance, Lenzerheide, location-scale parameterisation, London, MCMskv, Metropolis-Hastings algorithm, mixtures of distributions, non-informative priors, poster, R, reference priors, Switzerland, Ultimixt on January 11, 2016 by xi'an**A**fter presenting this work in both London and Lenzerheide, Kaniav Kamary, Kate Lee and I arXived and submitted our paper on a new parametrisation of location-scale mixtures. Although it took a long while to finalise the paper, given that we came with the original and central idea about a year ago, I remain quite excited by this new representation of mixtures, because the use of a global location-scale (hyper-)parameter doubling as the mean-standard deviation for the mixture itself implies that all the other parameters of this mixture model [beside the weights] belong to the intersection of a unit hypersphere with an hyperplane. [Hence the title above I regretted not using for the poster at MCMskv!]This realisation that using a (meaningful) hyperparameter (μ,σ) leads to a compact parameter space for the component parameters is important for inference in such mixture models in that the hyperparameter (μ,σ) is easily estimated from the entire sample, while the other parameters can be studied using a non-informative prior like the Uniform prior on the ensuing compact space. This non-informative prior for mixtures is something I have been seeking for many years, hence my on-going excitement! In the mid-1990‘s, we looked at a Russian doll type parametrisation with Kerrie Mengersen that used the “first” component as defining the location-scale reference for the entire mixture. And expressing each new component as a local perturbation of the previous one. While this is a similar idea than the current one, it falls short of leading to a natural non-informative prior, forcing us to devise a proper prior on the variance that was a mixture of a Uniform U(0,1) and of an inverse Uniform 1/U(0,1). Because of the lack of compactness of the parameter space. Here, fixing both mean and variance (or even just the variance) binds the mixture parameter to an ellipse conditional on the weights. A space that can be turned into the unit sphere via a natural reparameterisation. Furthermore, the intersection with the hyperplane leads to a closed form spherical reparameterisation. Yay!

While I do not wish to get into the debate about the [non-]existence of “non-informative” priors at this stage, I think being able to using the invariant reference prior π(μ,σ)=1/σ is quite neat here because the inference on the mixture parameters should be location and scale equivariant. The choice of the prior on the remaining parameters is of lesser importance, the Uniform over the compact being one example, although we did not study in depth this impact, being satisfied with the outputs produced from the default (Uniform) choice.

From a computational perspective, the new parametrisation can be easily turned into the old parametrisation, hence leads to a closed-form likelihood. This implies a Metropolis-within-Gibbs strategy can be easily implemented, as we did in the derived Ultimixt R package. (Which programming I was not involved in, solely suggesting the name *Ultimixt* from ultimate mixture parametrisation, a former title that we eventually dropped off for the paper.)

Discussing the paper at MCMskv was very helpful in that I got very positive feedback about the approach and superior arguments to justify the approach and its appeal. And to think about several extensions outside location scale families, if not in higher dimensions which remain a practical challenge (in the sense of designing a parametrisation of the covariance matrices in terms of the global covariance matrix).