a [counter]example of minimaxity

A chance question on X validated made me reconsider about the minimaxity over the weekend. Consider a Geometric G(p) variate X. What is the minimax estimator of p under squared error loss ? I thought it could be obtained via (Beta) conjugate priors, but following Dyubin (1978) the minimax estimator corresponds to a prior with point masses at ¼ and 1, resulting in a constant estimator equal to ¾ everywhere, except when X=0 where it is equal to 1. The actual question used a penalised qaudratic loss, dividing the squared error by p(1-p), which penalizes very strongly errors at p=0,1, and hence suggested an estimator equal to 1 when X=0 and to 0 otherwise. This proves to be the (unique) minimax estimator. With constant risk equal to 1. This reminded me of this fantastic 1984 paper by Georges Casella and Bill Strawderman on the estimation of the normal bounded mean, where the least favourable prior is supported by two atoms if the bound is small enough. Figure 1 in the Negative Binomial extension by Morozov and Syrova (2022) exploits the same principle. (Nothing Orwellian there!) If nothing else, a nice illustration for my Bayesian decision theory course!

principles of uncertainty (second edition)

A new edition of Principles of Uncertainty is about to appear. I was asked by CRC Press to review the new book and here are some (raw) extracts from my review. (Some comments may not apply to the final and published version, mind.)

In Chapter 6, the proof of the Central Limit Theorem utilises the “smudge” technique, which is to add an independent noise to both the sequence of rvs and its limit. This is most effective and reminds me of quite a similar proof Jacques Neveu used in its probability notes in Polytechnique. Which went under the more formal denomination of convolution, with the same (commendable) purpose of avoiding Fourier transforms. If anything, I would have favoured a slightly more condensed presentation in less than 8 pages. Is Corollary 6.5.8 useful or even correct??? I do not think so because the non-centred average rescaled by √n diverges almost surely. For the same reason, I object to the very first sentence of Section 6.5 (p.246)

In Chapter 7, I found a nice mention of (Hermann) Rubin’s insistence on not separating probability and utility as only the product matters. And another fascinating quote from Keynes, not from his early statistician’s years, but in 1937 as an established economist

“The sense in which I am using the term uncertain is that in which the prospect of a European war is uncertain, or the price of copper and the rate of interest twenty years hence, or the obsolescence of a new invention, or the position of private wealth-owners in the social system in 1970. About these matters there is no scientific basis on which to form any calculable probability whatever. We simply do not know. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook this awkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series of prospective advantages and disadvantages, each multiplied by its appropriate probability, waiting to the summed.”

(is the last sentence correct? I would have expected, pardon my French!, “to be summed”). Further interesting trivia on the criticisms of utility theory, including de Finetti’s role and his own lack of connection with subjective probability principles.

In Chapter 8, a major remark (iii) is found p.293 about the fact that a conjugate family requires a dominating measure (although this is expressed differently since the book shies away from introducing measure theory, ) reminds me of a conversation I had with Jay when I visited Carnegie Mellon in 2013 (?). Which exposes the futility of seeing conjugate priors as default priors. It is somewhat surprising that a notion like admissibility appears as a side quantity when discussing Stein’s paradox in 8.2.1 [and then later in Section 9.1.3] while it seems to me to be central to Bayesian decision theory, much more than the epiphenomenon that Stein’s paradox represents in the big picture. But the book dismisses minimaxity even faster in Section 9.1.4:

As many who suffer from paranoia have discovered, one can always dream-up an even worse possibility to guard against. Thus, the minimax framework is unstable. (p.336)

Interesting introduction of the Wishart distribution to kindly handle random matrices and matrix Jacobians, with the original space being the p(p+1)/2 real space (implicitly endowed with the Lebesgue measure). Rather than a more structured matricial space. A font error makes Corollary 8.7.2 abort abruptly. The space of positive definite matrices is mentioned in Section8.7.5 but still (implicitly) corresponds to the common p(p+1)/2 real Euclidean space. Another typo in Theorem 8.9.2 with a Frenchised version of Dirichlet, Dirichelet. Followed by a Dirchlet at the end of the proof (p.322). Again and again on p.324 and on following pages. I would object to the singular in the title of Section 8.10 as there are exponential families rather than a single one. With no mention made of Pitman-Koopman lemma and its consequences, namely that the existence of conjugacy remains an epiphenomenon. Hence making the amount of pages dedicated to gamma, Dirichlet and Wishart distributions somewhat excessive.

In Chapter 9, I noticed (p.334) a Scheffe that should be Scheffé (and again at least on p.444). (I love it that Jay also uses my favorite admissible (non-)estimator, namely the constant value estimator with value 3.) I wonder at the worth of a ten line section like 9.3, when there are delicate issues in handling meta-analysis, even in a Bayesian mood (or mode). In the model uncertainty section, Jay discuss the (im)pertinence of both selecting one of the models and setting independent priors on their respective parameters, with which I disagree on both levels. Although this is followed by a more reasonable (!) perspective on utility. Nice to see a section on causation, although I would have welcomed an insert on the recent and somewhat outrageous stand of Pearl (and MacKenzie) on statisticians missing the point on causation and counterfactuals by miles. Nonparametric Bayes is a new section, inspired from Ghahramani (2005). But while it mentions Gaussian and Dirichlet [invariably misspelled!] processes, I fear it comes short from enticing the reader to truly grasp the meaning of a prior on functions. Besides mentioning it exists, I am unsure of the utility of this section. This is one of the rare instances where measure theory is discussed, only to state this is beyond the scope of the book (p.349).

value of a chess game

In our (internal) webinar at CEREMADE today, Miguel Oliu Barton gave a talk on the recent result his student Luc Attia and himself obtained, namely a tractable way of finding the value of a game (when minimax equals maximin), result that got recently published in PNAS:

“Stochastic games were introduced by the Nobel Memorial Prize winner Lloyd Shapley in 1953 to model dynamic interactions in which the environment changes in response to the players’ behavior. The theory of stochastic games and its applications have been studied in several scientific disciplines, including economics, operations research, evolutionary biology, and computer science. In addition, mathematical tools that were used and developed in the study of stochastic games are used by mathematicians and computer scientists in other fields. This paper contributes to the theory of stochastic games by providing a tractable formula for the value of finite competitive stochastic games. This result settles a major open problem which remained unsolved for nearly 40 years.”

While I did not see a direct consequence of this result in regular statistics, I found most interesting the comment made at one point that chess (with forced nullity after repetitions) had a value, by virtue of Zermelo’s theorem. As I had never considered the question (contrary to Shannon!). This value remains unknown.

O’Bayes 19/3

Nancy Reid gave the first talk of the [Canada] day, in an impressive comparison of all approaches in statistics that involve a distribution of sorts on the parameter, connected with the presentation she gave at BFF4 in Harvard two years ago, including safe Bayes options this time. This was related to several (most?) of the talks at the conference, given the level of worry (!) about the choice of a prior distribution. But the main assessment of the methods still seemed to be centred on a frequentist notion of calibration, meaning that epistemic interpretations of probabilities and hence most of Bayesian answers were disqualified from the start.

In connection with Nancy’s focus, Peter Hoff’s talk also concentrated on frequency valid confidence intervals in (linear) hierarchical models. Using prior information or structure to build better and shrinkage-like confidence intervals at a given confidence level. But not in the decision-theoretic way adopted by George Casella, Bill Strawderman and others in the 1980’s. And also making me wonder at the relevance of contemplating a fixed coverage as a natural goal. Above, a side result shown by Peter that I did not know and which may prove useful for Monte Carlo simulation.

Jaeyong Lee worked on a complex model for banded matrices that starts with a regular Wishart prior on the unrestricted space of matrices, computes the posterior and then projects this distribution onto the constrained subspace. (There is a rather consequent literature on this subject, including works by David Dunson in the past decade of which I was unaware.) This is a smart demarginalisation idea but I wonder a wee bit at the notion as the constrained space has measure zero for the larger model. This could explain for the resulting posterior not being a true posterior for the constrained model in the sense that there is no prior over the constrained space that could return such a posterior. Another form of marginalisation paradox. The crux of the paper is however about constructing a functional form of minimaxity. In his discussion of the paper, Guido Consonni provided a representation of the post-processed posterior (P³) that involves the Dickey-Savage ratio, sort of, making me more convinced of the connection.

As a lighter aside, one item of local information I should definitely have broadcasted more loudly and long enough in advance to the conference participants is that the University of Warwick is not located in ye olde town of Warwick, where there is no university, but on the outskirts of the city of Coventry, but not to be confused with the University of Coventry. Located in Coventry.

 

distributed posteriors

Another presentation by our OxWaSP students introduced me to the notion of distributed posteriors, following a 2018 paper by Botond Szabó and Harry van Zanten. Which corresponds to the construction of posteriors when conducting a divide & conquer strategy. The authors show that an adaptation of the prior to the division of the sample is necessary to recover the (minimax) convergence rate obtained in the non-distributed case. This is somewhat annoying, except that the adaptation amounts to take the original prior to the power 1/m, when m is the number of divisions. They further show that when the regularity (parameter) of the model is unknown, the optimal rate cannot be recovered unless stronger assumptions are made on the non-zero parameters of the model.

“First of all, we show that depending on the communication budget, it might be advantageous to group local machines and let different groups work on different aspects of the high-dimensional object of interest. Secondly, we show that it is possible to have adaptation in communication restricted distributed settings, i.e. to have data-driven tuning that automatically achieves the correct bias-variance trade-off.”

I find the paper of considerable interest for scalable MCMC methods, even though the setting may happen to sound too formal, because the study incorporates parallel computing constraints. (Although I did not investigate the more theoretical aspects of the paper.)