## mathematical theory of Bayesian statistics [book review]

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , on May 6, 2021 by xi'an

I came by chance (and not by CHANCE) upon this 2018 CRC Press book by Sumio Watanabe and ordered it myself to gather which material it really covered. As the back-cover blurb was not particularly clear and the title sounded quite general. After reading it, I found out that this is a mathematical treatise on some aspects of Bayesian information criteria, in particular on the Widely Applicable Information Criterion (WAIC) that was introduced by the author in 2010. The result is a rather technical and highly focussed book with little motivation or intuition surrounding the mathematical results, which may make the reading arduous for readers. Some background on mathematical statistics and Bayesian inference is clearly preferable and the book cannot be used as a textbook for most audiences, as opposed to eg An Introduction to Bayesian Analysis by J.K. Ghosh et al. or even more to Principles of Uncertainty by J. Kadane. In connection with this remark the exercises found in the book are closer to the delivery of additional material than to textbook-style exercises.

“posterior distributions are often far from any normal distribution, showing that Bayesian estimation gives the more accurate inference than other estimation methods.”

The overall setting is one where both the sampling and the prior distributions are different from respective “true” distributions. Requiring a tool to assess the discrepancy when utilising a specific pair of such distributions. Especially when the posterior distribution cannot be approximated by a Normal distribution. (Lindley’s paradox makes an interesting incognito incursion on p.238.) The WAIC is supported for the determination of the “true” model, in opposition to AIC and DIC, incl. on a mixture example that reminded me of our eight versions of DIC paper. In the “Basic Bayesian Theory” chapter (§3), the “basic theorem of Bayesian statistics” (p.85) states that the various losses related with WAIC can be expressed as second-order Taylor expansions of some cumulant generating functions, with order o(n⁻¹), “even if the posterior distribution cannot be approximated by any normal distribution” (p.87). With the intuition that

“if a log density ratio function has a relatively finite variance then the generalization loss, the cross validation loss, the training loss and WAIC have the same asymptotic behaviors.”

Obviously, these “basic” aspects should come as a surprise to a fair percentage of Bayesians (in the sense of not being particularly basic). Myself included. Chapter 4 exposes why, for regular models, the posterior distribution accumulates in an ε neighbourhood of the optimal parameter at a speed O(n2/5). With the normalised partition function being of order n-d/2 in the neighbourhood and exponentially negligible outside. A consequence of this regular asymptotic theory is that all above losses are asymptotically equivalent to the negative log likelihood plus similar order n⁻¹ terms that can be ordered. Chapters 5 and 6 deal with “standard” [the likelihood ratio is a multi-index power of the parameter ω] and general posterior distributions that can be written as mixtures of standard distributions,  with expressions of the above losses in terms of new universal constants. Again, a rather remote concern of mine. The book also includes a chapter (§7) on MCMC, with a rather involved proof that a Metropolis algorithm satisfies detailed balance (p.210). The Gibbs sampling section contains an extensive example on a two-dimensional two-component unit-variance Normal mixture, with an unusual perspective on the posterior, which is considered as “singular” when the true means are close. (Label switching or the absence thereof is not mentioned.) In terms of approximating the normalising constant (or free energy), the only method discussed there is path sampling, with a cryptic remark about harmonic mean estimators (not identified as such). In a final knapsack chapter (§9),  Bayes factors (confusedly denoted as L(x)) are shown to be most powerful tests in a Bayesian sense when comparing hypotheses without prior weights on said hypotheses, while posterior probability ratios are the natural statistics for comparing models with prior weights on said models. (With Lindley’s paradox making another appearance, still incognito!) And a  notion of phase transition for hyperparameters is introduced, with the meaning of a radical change of behaviour at a critical value of said hyperparameter. For instance, for a simple normal- mixture outlier model, the critical value of the Beta hyperparameter is α=2. Which is a wee bit of a surprise when considering Rousseau and Mengersen (2011) since their bound for consistency was α=d/2.

In conclusion, this is quite an original perspective on Bayesian models, covering the somewhat unusual (and potentially controversial) issue of misspecified priors and centered on the use of information criteria. I find the book could have benefited from further editing as I noticed many typos and somewhat unusual sentences (at least unusual to me).

[Disclaimer about potential self-plagiarism: this post or an edited version should eventually appear in my Books Review section in CHANCE.]

## Lindley’s paradox as a loss of resolution

Posted in Books, pictures, Statistics with tags , , , , , , , , on November 9, 2016 by xi'an

“The principle of indifference states that in the absence of prior information, all mutually exclusive models should be assigned equal prior probability.”

Colin LaMont and Paul Wiggins arxived a paper on Lindley’s paradox a few days ago. The above quote is the (standard) argument for picking (½,½) partition between the two hypotheses, which I object to if only because it does not stand for multiple embedded models. The main point in the paper is to argue about the loss of resolution induced by averaging against the prior, as illustrated by the picture above for the N(0,1) versus N(μ,1) toy problem. What they call resolution is the lowest possible mean estimate for which the null is rejected by the Bayes factor (assuming a rejection for Bayes factors larger than 1). While the detail is missing, I presume the different curves on the lower panel correspond to different choices of L when using U(-L,L) priors on μ… The “Bayesian rejoinder” to the Lindley-Bartlett paradox (p.4) is in tune with my interpretation, namely that as the prior mass under the alternative gets more and more spread out, there is less and less prior support for reasonable values of the parameter, hence a growing tendency to accept the null. This is an illustration of the long-lasting impact of the prior on the posterior probability of the model, because the data cannot impact the tails very much.

“If the true prior is known, Bayesian inference using the true prior is optimal.”

This sentence and the arguments following is meaningless in my opinion as knowing the “true” prior makes the Bayesian debate superfluous. If there was a unique, Nature provided, known prior π, it would loose its original meaning to become part of the (frequentist) model. The argument is actually mostly used in negative, namely that since it is not know we should not follow a Bayesian approach: this is, e.g., the main criticism in Inferential Models. But there is no such thing as a “true” prior! (Or a “true’ model, all things considered!) In the current paper, this pseudo-natural approach to priors is utilised to justify a return to the pseudo-Bayes factors of the 1990’s, when one part of the data is used to stabilise and proper-ise the (improper) prior, and a second part to run the test per se. This includes an interesting insight on the limiting cases of partitioning corresponding to AIC and BIC, respectively, that I had not seen before. With the surprising conclusion that “AIC is the derivative of BIC”!

## model selection and multiple testing

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on October 23, 2015 by xi'an

Ritabrata Dutta, Malgorzata Bogdan and Jayanta Ghosh recently arXived a survey paper on model selection and multiple testing. Which provides a good opportunity to reflect upon traditional Bayesian approaches to model choice. And potential alternatives. On my way back from Madrid, where I got a bit distracted when flying over the South-West French coast, from Biarritz to Bordeaux. Spotting the lake of Hourtain, where I spent my military training month, 29 years ago!

“On the basis of comparison of AIC and BIC, we suggest tentatively that model selection rules should be used for the purpose for which they were introduced. If they are used for other problems, a fresh justification is desirable. In one case, justification may take the form of a consistency theorem, in the other some sort of oracle inequality. Both may be hard to prove. Then one should have substantial numerical assessment over many different examples.”

The authors quickly replace the Bayes factor with BIC, because it is typically consistent. In the comparison between AIC and BIC they mention the connundrum of defining a prior on a nested model from the prior on the nesting model, a problem that has not been properly solved in my opinion. The above quote with its call to a large simulation study reminded me of the paper by Arnold & Loeppky about running such studies through ecdfs. That I did not see as solving the issue. The authors also discuss DIC and Lasso, without making much of a connection between those, or with the above. And then reach the parametric empirical Bayes approach to model selection exemplified by Ed George’s and Don Foster’s 2000 paper. Which achieves asymptotic optimality for posterior prediction loss (p.9). And which unifies a wide range of model selection approaches.

A second part of the survey considers the large p setting, where BIC is not a good approximation to the Bayes factor (when testing whether or not all mean entries are zero). And recalls that there are priors ensuring consistency for the Bayes factor in this very [restrictive] case. Then, in Section 4, the authors move to what they call “cross-validatory Bayes factors”, also known as partial Bayes factors and pseudo-Bayes factors, where the data is split to (a) make the improper prior proper and (b) run the comparison or test on the remaining data. They also show the surprising result that, provided the fraction of the data used to proper-ise the prior does not converge to one, the X validated Bayes factor remains consistent [for the special case above]. The last part of the paper concentrates on multiple testing but is more tentative and conjecturing about convergence results, centring on the differences between full Bayes and empirical Bayes. Then the plane landed in Paris and I stopped my reading, not feeling differently about the topic than when the plane started from Madrid.

## a unified treatment of predictive model comparison

Posted in Books, Statistics, University life with tags , , , , , , , , , on June 16, 2015 by xi'an

“Applying various approximation strategies to the relative predictive performance derived from predictive distributions in frequentist and Bayesian inference yields many of the model comparison techniques ubiquitous in practice, from predictive log loss cross validation to the Bayesian evidence and Bayesian information criteria.”

Michael Betancourt (Warwick) just arXived a paper formalising predictive model comparison in an almost Bourbakian sense! Meaning that he adopts therein a very general representation of the issue, with minimal assumptions on the data generating process (excluding a specific metric and obviously the choice of a testing statistic). He opts for an M-open perspective, meaning that this generating process stands outside the hypothetical statistical model or, in Lindley’s terms, a small world. Within this paradigm, the only way to assess the fit of a model seems to be through the predictive performances of that model. Using for instance an f-divergence like the Kullback-Leibler divergence, based on the true generated process as the reference. I think this however puts a restriction on the choice of small worlds as the probability measure on that small world has to be absolutely continuous wrt the true data generating process for the distance to be finite. While there are arguments in favour of absolutely continuous small worlds, this assumes a knowledge about the true process that we simply cannot gather. Ignoring this difficulty, a relative Kullback-Leibler divergence can be defined in terms of an almost arbitrary reference measure. But as it still relies on the true measure, its evaluation proceeds via cross-validation “tricks” like jackknife and bootstrap. However, on the Bayesian side, using the prior predictive links the Kullback-Leibler divergence with the marginal likelihood. And Michael argues further that the posterior predictive can be seen as the unifying tool behind information criteria like DIC and WAIC (widely applicable information criterion). Which does not convince me towards the utility of those criteria as model selection tools, as there is too much freedom in the way approximations are used and a potential for using the data several times.

## An objective prior that unifies objective Bayes and information-based inference

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , on June 8, 2015 by xi'an

During the Valencia O’Bayes 2015 meeting, Colin LaMont and Paul Wiggins arxived a paper entitled “An objective prior that unifies objective Bayes and information-based inference”. It would have been interesting to have the authors in Valencia, as they make bold claims about their w-prior as being uniformly and maximally uninformative. Plus achieving this unification advertised in the title of the paper. Meaning that the free energy (log transform of the inverse evidence) is the Akaike information criterion.

The paper starts by defining a true prior distribution (presumably in analogy with the true value of the parameter?) and generalised posterior distributions as associated with any arbitrary prior. (Some notations are imprecise, check (3) with the wrong denominator or the predictivity that is supposed to cover N new observations on p.2…) It then introduces a discretisation by considering all models within a certain Kullback divergence δ to be undistinguishable. (A definition that does not account for the assymmetry of the Kullback divergence.) From there, it most surprisingly [given the above discretisation] derives a density on the whole parameter space

$\pi(\theta) \propto \text{det} I(\theta)^{1/2} (N/2\pi \delta)^{K/2}$

where N is the number of observations and K the dimension of θ. Dimension which may vary. The dependence on N of the above is a result of using the predictive on N points instead of one. The w-prior is however defined differently: “as the density of indistinguishable models such that the multiplicity is unity for all true models”. Where the log transform of the multiplicity is the expected log marginal likelihood minus the expected log predictive [all expectations under the sampling distributions, conditional on θ]. Rather puzzling in that it involves the “true” value of the parameter—another notational imprecision, since it has to hold for all θ’s—as well as possibly improper priors. When the prior is improper, the log-multiplicity is a difference of two terms such that the first term depends on the constant used with the improper prior, while the second one does not…  Unless the multiplicity constraint also determines the normalising constant?! But this does not seem to be the case when considering the following section on normalising the w-prior. Mentioning a “cutoff” for the integration that seems to pop out of nowhere. Curiouser and curiouser. Due to this unclear handling of infinite mass priors, and since the claimed properties of uniform and maximal uninformativeness are not established in any formal way, and since the existence of a non-asymptotic solution to the multiplicity equation is neither demonstrated, I quickly lost interest in the paper. Which does not contain any worked out example. Read at your own risk!