robust privacy

During a recent working session, some Oceanerc (incl. me) went reading Privacy-Preserving Parametric Inference: A Case for Robust Statistics by Marco Avella-Medina (JASA, 2022), where robust criteria are advanced as efficient statistical tools in private settings. In this paper, robustness means using M-estimators T—as function of the empirical cdf—with basis score functions Ψ, defined as

where Ψ is bounded. A construction further requiring that one can assess the sensitivity (in Dwork et al, 2006, sense) of a queried function, sensitivity itself linked with a measure of differential privacy. Because standard robustness approaches à la Huber allow for a portion of the sample to issue from an outlying (arbitrary) distribution, as in ε-contaminations, it makes perfect sense that robustness emerges within the differential framework. However, this common sense perception does not seem good enough for achieving differential privacy and the paper introduces a further randomization with noise scaled by (n,ε,δ) in the following way

that also applies to test statistics. This scaling seems to constitute the central result of the paper, which establishes asymptotically validity in the sense of statistical consistency (with the sample size n). But I am left wondering whether this outcome counts as supporting differential privacy as a sensible notion…

“…our proofs for the convergence of noisy gradient descent and noisy Newton’s method rely on showing that with high probability, the noise introduced to the gradients and Hessians has a negligible effect on the convergence of the iterates (up to the order of the statistical error of the non-noisy versions of the algorithms).” Avella-Medina, Bradshaw, & Loh

As a sequel I then read a more recent publication of Avella-Medina, Differentially private inference via noisy optimization, written with Casey Bradshaw & Po-Ling Loh, which appeared in the Annals of Statistics (2023). Again considering privatised estimation and inference for M-estimators, obtained by using noisy optimization procedures (noisy gradient descent, noisy Newton’s method) and constructing noisy confidence regions, that output differentially private avatars of standard M-estimators. Here the noisification goes through a randomisation of the gradient step like

where B is an upper bound on the gradient Ψ, η is a discretization step, and K is the total number of iterations (thus fixed in advance). The above stochastic gradient sequence converges with high probability to the actual M-estimator in n and not in K, since the upper bound on the distance scales in √K/n. Where does the attached privacy guarantee come from? It proceeds by an argument of a composition of a sequence of differentially private outputs, all based on the same dataset.

“…the larger the number [K] of data (gradient) queries of the algorithm, the more prone it will be to privacy leakage.”

The Newton method version is a variation on the above stochastic gradient descent. Except it seems to converge faster, as illustrated above.

statistical modeling with R [book review]

Statistical Modeling with R (A dual frequentist and Bayesian approach for life scientists) is a recent book written by Pablo Inchausti, from Uruguay. In a highly personal and congenial style (witness the preface), with references to (fiction) books that enticed me to buy them. The book was sent to me by the JASA book editor for review and I went through the whole of it during my flight back from Jeddah. [Disclaimer about potential self-plagiarism: this post or a likely edited version of it will eventually appear in JASA. If not CHANCE, for once.]

The very first sentence (after the preface) quotes my late friend Steve Fienberg, which is definitely starting on the right foot. The exposition of the motivations for writing the book is quite convincing, with more emphasis than usual put on the notion and limitations of modeling. The discourse is overall inspirational and contains many relevant remarks and links that make it worth reading it as a whole. While heavily connected with a few R packages like fitdist, fitistrplus, brms (a  front for Stan), glm, glmer, the book is wisely bypassing the perilous reef of recalling R bases. Similarly for the foundations of probability and statistics. While lacking in formal definitions, in my opinion, it reads well enough to somehow compensate for this very lack. I also appreciate the coherent and throughout continuation of the parallel description of Bayesian and non-Bayesian analyses, an attempt that often too often quickly disappear in other books. (As an aside, note that hardly anyone claims to be a frequentist, except maybe Deborah Mayo.) A new model is almost invariably backed by a new dataset, if a few being somewhat inappropriate as in the mammal sleep patterns of Chapter 5. Or in Fig. 6.1.

Given that the main motivation for the book (when compared with references like BDA) is heavily towards the practical implementation of statistical modelling via R packages, it is inevitable that a large fraction of Statistical Modeling with R is spent on the analysis of R outputs, even though it sometimes feels a wee bit too heavy for yours truly.  The R screen-copies are however produced in moderate quantity and size, even though the variations in typography/fonts (at least on my copy?!) may prove confusing. Obviously the high (explosive?) distinction between regression models may eventually prove challenging for the novice reader. The specific issue of prior input (or “defining priors”) is briefly addressed in a non-chapter (p.323), although mentions are made throughout preceding chapters. I note the nice appearance of hierarchical models and experimental designs towards the end, but would have appreciated some discussions on missing topics such as time series, causality, connections with machine learning, non-parametrics, model misspecification. As an aside, I appreciated being reminded about the apocryphal nature of Ockham’s much cited quote “Pluralitas non est ponenda sine necessitate“.

Typo Jeffries found in Fig. 2.1, along with a rather sketchy representation of the history of both frequentist and Bayesian statistics. And Jon Wakefield’s book (with related purpose of presenting both versions of parametric inference) was mistakenly entered as Wakenfield’s in the bibliography file. Some repetitions occur. I do not like the use of the equivalence symbol ≈ for proportionality. And I found two occurrences of the unavoidable “the the” typo (p.174 and p.422). I also had trouble with some sentences like “long-run, hypothetical distribution of parameter estimates known as the sampling distribution” (p.27), “maximum likelihood estimates [being] sufficient” (p.28), “Jeffreys’ (1939) conjugate priors” [which were introduced by Raiffa and Schlaifer] (p.35), “A posteriori tests in frequentist models” (p.130), “exponential families [having] limited practical implications for non-statisticians” (p.190), “choice of priors being correct” (p.339), or calling MCMC sample terms “estimates” (p.42), and issues with some repetitions, missing indices for acronyms, packages, datasets, but did not bemoan the lack homework sections (beyond suggesting new datasets for analysis).

A problematic MCMC entry is found when calibrating the choice of the Metropolis-Hastings proposal towards avoiding negative values “that will generate an error when calculating the log-likelihood” (p.43) since it suggests proposed values should not exceed the support of the posterior (and indicates a poor coding of the log-likelihood!). I also find the motivation for the full conditional decomposition behind the Gibbs sampler (p.47) unnecessarily confusing. (And automatically having a Metropolis-Hastings step within Gibbs as on Fig. 3.9 brings another magnitude of confusion.) The Bayes factor section is very terse. The derivation of the Kullback-Leibler representation (7.3) as an expected log likelihood ratio seems to be missing a reference measure. Of course, seeing a detailed coverage of DIC (Section 7.4) did not suit me either, even though the issue with mixtures was alluded to (with no detail whatsoever). The Nelder presentation of the generalised linear models felt somewhat antiquated, since the addition of the scale factor a(φ) sounds over-parameterized.

But those are minor quibble in relation to a book that should attract curious minds of various background knowledge and expertise in statistics, as well as work nicely to support an enthusiastic teacher of statistical modelling. I thus recommend this book most enthusiastically.

identifying mixtures

I had not read this 2017 discussion of Bayesian mixture estimation by Michael Betancourt before I found it mentioned in a recent paper. Where he re-explores the issue of identifiability and label switching in finite mixture models. Calling somewhat abusively degenerate mixtures where all components share the same family, e.g., mixtures of Gaussians. Illustrated by Stan code and output. This is rather traditional material, in that the non-identifiability of mixture components has been discussed in many papers and at least as many solutions proposed to overcome the difficulties of exploring the posterior distribution. Including our 2000 JASA paper with Gilles Celeux and Merrilee Hurn. With my favourite approach being the label-free representations as a point process in the parameter space (following an idea of Peter Green) or as a collection of clusters in the latent variable space. I am much less convinced by ordering constraints: while they formally differentiate and therefore identify the individual components of a mixture, they partition the parameter space with no regard towards the geometry of the posterior distribution. With in turn potential consequences on MCMC explorations of this fragmented surface that creates barriers for simulated Markov chains. Plus further difficulties with inferior but attracting modes in identifiable situations.

ordered allocation sampler

Recently, Pierpaolo De Blasi and María Gil-Leyva arXived a proposal for a novel Gibbs sampler for mixture models. In both finite and infinite mixture models. In connection with Pitman (1996) theory of species sampling and with interesting features in terms of removing the vexing label switching features.

“The key idea is to work with the mixture components in the random order of appearance in an exchangeable sequence from the mixing distribution (…) In accordance with the order of appearance, we derive a new Gibbs sampling algorithm that we name the ordered allocation sampler. “

This central idea is thus a reinterpretation of the mixture model as the marginal of the component model when its parameter is distributed as a species sampling variate. An ensuing marginal algorithm is to integrate out the weights and the allocation variables to only consider the non-empty component parameters and the partition function, which are label invariant. Which reminded me of the proposal we made in our 2000 JASA paper with Gilles Celeux and Merrilee Hurn (one of my favourite papers!). And of the [first paper in Statistical Methodology] 2004 partitioned importance sampling version with George Casella and Marty Wells. As in the later, the solution seems to require the prior on the component parameters to be conjugate (as I do not see a way to produce an unbiased estimator of the partition allocation probabilities).

The ordered allocation sample considers the posterior distribution of the different object made of the parameters and of the sequence of allocations to the components for the sample written in a given order, ie y¹,y², &tc. Hence y¹ always gets associated with component 1, y² with either component 1 or component 2, and so on. For this distribution, the full conditionals are available, incl. the full posterior on the number m of components, only depending on the data through the partition sizes and the number m⁺ of non-empty components. (Which relates to the debate as to whether or not m is estimable…) This sequential allocation reminded me as well of an earlier 2007 JRSS paper by Nicolas Chopin. Albeit using particles rather than Gibbs and applied to a hidden Markov model. Funny enough, their synthetic dataset univ4 almost resembles the Galaxy dataset (as in the above picture of mine)!

Handbooks [not a book review]