connection between tempering & entropic mirror descent

The next One World ABC webinar is this  Thursday,  the 2nd May, at 9am UK time, with Francesca Crucinio (King’s College London, formerly CREST and even more formerly Warwick) presenting

“A connection between Tempering and Entropic Mirror Descent”.

a joint work with Nicolas Chopin and Anna Korba (both from CREST) whose abstract follows:

This work explores the connections between tempering (for Sequential Monte Carlo; SMC) and entropic mirror descent to sample from a target probability distribution whose unnormalized density is known. We establish that tempering SMC corresponds to entropic mirror descent applied to the reverse Kullback-Leibler (KL) divergence and obtain convergence rates for the tempering iterates. Our result motivates the tempering iterates from an optimization point of view, showing that tempering can be seen as a descent scheme of the KL divergence with respect to the Fisher-Rao geometry, in contrast to Langevin dynamics that perform descent of the KL with respect to the Wasserstein-2 geometry. We exploit the connection between tempering and mirror descent iterates to justify common practices in SMC and derive adaptive tempering rules that improve over other alternative benchmarks in the literature.

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.]