politics coming [too close to] statistics [or the reverse]

On 30 April, David Spiegelhalter wrote an opinion column in The Guardian, Coronavirus deaths: how does Britain compare with other countries?, where he pointed out the difficulty, even “for a bean-counting statistician to count deaths”, as the reported figures are undercounts, and stated that “many feel that excess deaths give a truer picture of the impact of an epidemic“. Which, on the side, I indeed believe is a more objective material, as also reported by INSEE and INED in France.

“…my cold, statistical approach is to wait until the end of the year, and the years after that, when we can count the excess deaths. Until then, this grim contest won’t produce any league tables we can rely on.” D. Spiegelhalter

My understanding of the tribune is that the quick accumulation of raw numbers, even for deaths, and their use in the comparison of procedures and countries is not helping in understanding the impacts of policies and actions-reactions from a week ago. Starting with the delays in reporting death certificates, as again illustrated by the ten day lag in the INSEE reports. And accounting for covariates such as population density, economic and health indicators. (The graph below for instance relies on deaths so far attributed to COVID-19 rather than on excess deaths, while these attributions depend on the country policy and its official statistics capacities.)

“Polite request to PM and others: please stop using my Guardian article to claim we cannot make any international comparisons yet. I refer only to detailed league tables—of course we should now use other countries to try and learn why our numbers are high.” D. Spiegelhalter

However, when on 6 May Boris Johnson used this Guardian article during prime minister’s questions in the UK Parliement, to defuse a question from the Labour leader, Keir Starmer, David Spiegelhalter reacted with the above tweet, which is indeed that even with poor and undercounted data the total number of cases is much worse than predicted by the earlier models and deadlier than in neighbouring countries. Anyway, three other fellow statisticians, Phil Brown, Jim Smith (Warwick), and Henry Wynn, also reacted to David’s tribune by complaining at the lack of statistical modelling behind it and the fatalistic message it carries, advocating for model based decision-making, which would be fine if the data was not so unreliable… or if the proposed models were equipped with uncertainty bumpers accounting for misspecification and erroneous data.

a generalized representation of Bayesian inference

Jeremias Knoblauch, Jack Jewson and Theodoros Damoulas, all affiliated with Warwick (hence a potentially biased reading!), arXived a paper on loss-based Bayesian inference that Jack discussed with me on my last visit to Warwick. As I was somewhat scared by the 61 pages, of which the 8 first pages are in NeurIPS style. The authors argue for a decision-theoretic approach to Bayesian inference that involves a loss over distributions and a divergence from the prior. For instance, when using the log-score as the loss and the Kullback-Leibler divergence, the regular posterior emerges, as shown by Arnold Zellner. Variational inference also falls under this hat. The argument for this generalization is that any form of loss can be used and still returns a distribution that is used to assess uncertainty about the parameter (of interest). In the axioms they produce for justifying the derivation of the optimal procedure, including cases where the posterior is restricted to a certain class, one [Axiom 4] generalizes the likelihood principle. Given the freedom brought by this general framework, plenty of fringe Bayes methods like standard variational Bayes can be seen as solutions to such a decision problem. Others like EP do not. Of interest to me are the potentials for this formal framework to encompass misspecification and likelihood-free settings, as well as for assessing priors, which is always a fishy issue. (The authors mention in addition the capacity to build related specific design Bayesian deep networks, of which I know nothing.) The obvious reaction of mine is one of facing an abundance of wealth (!) but encompassing approximate Bayesian solutions within a Bayesian framework remains an exciting prospect.

O’Bayes 19/1 [snapshots]

Although the tutorials of O’Bayes 2019 of yesterday were poorly attended, albeit them being great entries into objective Bayesian model choice, recent advances in MCMC methodology, and the multiple layers of BART, for which I have to blame myself for sticking the beginning of O’Bayes too closely to the end of BNP as only the most dedicated could achieve the commuting from Oxford to Coventry to reach Warwick in time, the first day of talks were well attended, despite weekend commitments, conference fatigue, and perfect summer weather! Here are some snapshots from my bench (and apologies for not covering better the more theoretical talks I had trouble to follow, due to an early and intense morning swimming lesson! Like Steve Walker’s utility based derivation of priors that generalise maximum entropy priors. But being entirely independent from the model does not sound to me like such a desirable feature… And Natalia Bochkina’s Bernstein-von Mises theorem for a location scale semi-parametric model, including a clever construct of a mixture of two Dirichlet priors to achieve proper convergence.)

Jim Berger started the day with a talk on imprecise probabilities, involving the society for imprecise probability, which I discovered while reading Keynes’ book, with a neat resolution of the Jeffreys-Lindley paradox, when re-expressing the null as an imprecise null, with the posterior of the null no longer converging to one, with a limit depending on the prior modelling, if involving a prior on the bias as well, with Chris discussing the talk and mentioning a recent work with Edwin Fong on reinterpreting marginal likelihood as exhaustive X validation, summing over all possible subsets of the data [using log marginal predictive].Håvard Rue did a follow-up talk from his Valencià O’Bayes 2015 talk on PC-priors. With a pretty hilarious introduction on his difficulties with constructing priors and counseling students about their Bayesian modelling. With a list of principles and desiderata to define a reference prior. However, I somewhat disagree with his argument that the Kullback-Leibler distance from the simpler (base) model cannot be scaled, as it is essentially a log-likelihood. And it feels like multivariate parameters need some sort of separability to define distance(s) to the base model since the distance somewhat summarises the whole departure from the simpler model. (Håvard also joined my achievement of putting an ostrich in a slide!) In his discussion, Robin Ryder made a very pragmatic recap on the difficulties with constructing priors. And pointing out a natural link with ABC (which brings us back to Don Rubin’s motivation for introducing the algorithm as a formal thought experiment).

Sara Wade gave the final talk on the day about her work on Bayesian cluster analysis. Which discussion in Bayesian Analysis I alas missed. Cluster estimation, as mentioned frequently on this blog, is a rather frustrating challenge despite the simple formulation of the problem. (And I will not mention Larry’s tequila analogy!) The current approach is based on loss functions directly addressing the clustering aspect, integrating out the parameters. Which produces the interesting notion of neighbourhoods of partitions and hence credible balls in the space of partitions. It still remains unclear to me that cluster estimation is at all achievable, since the partition space explodes with the sample size and hence makes the most probable cluster more and more unlikely in that space. Somewhat paradoxically, the paper concludes that estimating the cluster produces a more reliable estimator on the number of clusters than looking at the marginal distribution on this number. In her discussion, Clara Grazian also pointed the ambivalent use of clustering, where the intended meaning somehow diverges from the meaning induced by the mixture model.

leave Bayes factors where they once belonged

In the past weeks I have received and read several papers (and X validated entries)where the Bayes factor is used to compare priors. Which does not look right to me, not on the basis of my general dislike of Bayes factors!, but simply because this seems to clash with the (my?) concept of Bayesian model choice and also because data should not play a role in that situation, from being used to select a prior, hence at least twice to run the inference, to resort to a single parameter value (namely the one behind the data) to decide between two distributions, to having no asymptotic justification, to eventually favouring the prior concentrated on the maximum likelihood estimator. And more. But I fear that this reticence to test for prior adequacy also extends to the prior predictive, or Box’s p-value, namely the probability under this prior predictive to observe something “more extreme” than the current observation, to quote from David Spiegelhalter.

let the evidence speak [book review]

This book by Alan Jessop, professor at the Durham University Business School,  aims at presenting Bayesian ideas and methods towards decision making “without formula because they are not necessary; the ability to add and multiply is all that is needed.” The trick is in using a Bayes grid, in other words a two by two table. (There are a few formulas that survived the slaughter, see e.g. on p. 91 the formula for the entropy. Contained in the chapter on information that I find definitely unclear.) When leaving the 2×2 world, things become more complicated and the construction of a prior belief as a probability density gets heroic without the availability of maths formulas. The first part of the paper is about Likelihood, albeit not the likelihood function, despite having the general rule that (p.73)

belief is proportional to base rate x likelihood

which is the book‘s version of Bayes’ (base?!) theorem. It then goes on to discuss the less structure nature of prior (or prior beliefs) against likelihood by describing Tony O’Hagan’s way of scaling experts’ beliefs in terms of a Beta distribution. And mentioning Jaynes’ maximum entropy prior without a single formula. What is hard to fathom from the text is how can one derive the likelihood outside surveys. (Using the illustration of 1963 Oswald’s murder by Ruby in the likelihood chapter does not particularly help!) A bit of nitpicking at this stage: the sentence

“The ancient Greeks, and before them the Chinese and the Aztecs…”

is historically incorrect since, while the Chinese empire dates back before the Greek dark ages, the Aztecs only rule Mexico from the 14th century (AD) until the Spaniard invasion. While most of the book sticks with unidimensional parameters, it also discusses more complex structures, for which it relies on Monte Carlo, although the description is rather cryptic (use your spreadsheet!, p.133). The book at this stage turns into a more story-telling mode, by considering for instance the Federalist papers analysis by Mosteller and Wallace. The reader can only follow the process of assessing a document authorship for a single word, as multidimensional cases (for either data or parameters) are out of reach. The same comment applies to the ecology, archeology, and psychology chapters that follow. The intermediary chapter on the “grossly misleading” [Court wording] of the statistical evidence in the Sally Clark prosecution is more accessible in that (again) it relies on a single number. Returning to the ban of Bayes rule in British courts:

In the light of the strong criticism by this court in the 1990s of using Bayes theorem before the jury in cases where there was no reliable statistical evidence, the practice of using a Bayesian approach and likelihood ratios to formulate opinions placed before a jury without that process being disclosed and debated in court is contrary to principles of open justice.

the discussion found in the book is quite moderate and inclusive, in that a Bayesian analysis helps in gathering evidence about a case, but may be misunderstood or misused at the [non-Bayesian] decision level.

In conclusion, Let the Evidence Speak is an interesting introduction to Bayesian thinking, through a simplifying device, the Bayes grid, which seems to come from management, with a large number of examples, if not necessarily all realistic and some side-stories. I doubt this exposure can produce expert practitioners, but it makes for an worthwhile awakening for someone “likely to have read this book because [one] had heard of Bayes but were uncertain what is was” (p.222). With commendable caution and warnings along the way.