ABC with privacy

I very recently read a  2021 paper by Mijung Park, Margarita Vinaroz, and Wittawat Jitkrittum on running ABC while ensuring data privacy (published in Entropy).

“…adding noise to the distance computed on the real observations and pseudo-data suffices the privacy guarantee of the resulting  posterior samples”

For ABC tolerance, they use maximum mean discrepancy (MMD) and for privacy the standard if unconvincing notion of differential privacy, defined by ensuring an upper bound on the amount of variation in the probability ratio when replacing/removing/adding an observation. (But not clearly convincing users their data is secure.)

While I have no reservation about the validation of the double-noise approach, I find it surprising that noise must be (twice) added when vanilla ABC is already (i) noisy, since based on random pseudo-data, and (ii) producing only a sample from an approximate posterior instead of returning an exact posterior. My impression indeed was that ABC should be good enough by itself to achieve privacy protection. In the sense that the accepted parameter values were those that generated random samples sufficiently close to the actual data, hence not only compatible with the true data, but also producing artificial datasets that are close enough to the data. Presumably these artificial datasets should not be produced as the intersection of their ε neighbourhoods may prove enough to identify the actual data. (The proposed algorithm does return all generated datasets.) Instead the supported algorithm involves randomisation of both tolerance ε and distance ρ to the observed data (with the side issue that they may become negative since the noise is Laplace).

approximate Bayesian inference [survey]

In connection with the special issue of Entropy I mentioned a while ago, Pierre Alquier (formerly of CREST) has written an introduction to the topic of approximate Bayesian inference that is worth advertising (and freely-available as well). Its reference list is particularly relevant. (The deadline for submissions is 21 June,)

special issue of Entropy

EntropyMCMC [R package]

My colleague from the Université d’Orléans, Didier Chauveau, has just published on CRAN a new R package called EntropyMCMC, which contains convergence assessment tools for MCMC algorithms, based on non-parametric estimates of the Kullback-Leibler divergence between current distribution and target. (A while ago, quite a while ago!, we actually collaborated with a few others on the Springer-Verlag Lecture Note #135 Discretization and MCMC convergence assessments.) This follows from a series of papers by Didier Chauveau and Pierre Vandekerkhove that started with a nearest neighbour entropy estimate. The evaluation of this entropy is based on N iid (parallel) chains, which involves a parallel implementation. While the missing normalising constant is overwhelmingly unknown, the authors this is not a major issue “since we are mostly interested in the stabilization” of the entropy distance. Or in the comparison of two MCMC algorithms. [Disclaimer: I have not experimented with the package so far, hence cannot vouch for its performances over large dimensions or problematic targets, but would as usual welcome comments and feedback on readers’ experiences.]

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.