David Dunson (Duke University)

Robust and scalable Bayes via the median posterior

Bayesian methods have great promise in big data sets, but this promise has not been fully realized due to the lack of scalable computational methods. Usual MCMC and SMC algorithms bog down as the size of the data and number of parameters increase. For massive data sets, it has become routine to rely on penalized optimization approaches implemented on distributed computing systems. The most popular scalable approximation algorithms rely on variational Bayes, which lacks theoretical guarantees and badly under-estimates posterior covariance. Another problem with Bayesian inference is the lack of robustness; data contamination and corruption is particularly common in large data applications and cannot easily be dealt with using traditional methods. We propose to solve both the robustness and the scalability problem using a new alternative to exact Bayesian inference we refer to as the median posterior. Data are divided into subsets and stored on different computers prior to analysis. For each subset, we obtain a stochastic approximation to the full data posterior, and run MCMC to generate samples from this approximation. The median posterior is defined as the geometric median of the subset-specific approximations, and can be rapidly approximated. We show several strong theoretical results for the median posterior, including general theorems on concentration rates and robustness. The methods are illustrated through simple examples, including Gaussian process regression with outliers.

Eric Moulines (Télécom ParisTech)

Proximal Metropolis adjusted Langevin algorithm for sampling sparse distribution over high-dimensional spaces

This talk introduces a new Markov Chain Monte Carlo method to sampling sparse distributions or to perform Bayesian model choices in high dimensional settings. The algorithm is a Hastings-Metropolis sampler with a proposal mechanism which combines (i) a Metropolis adjusted Langevin step to propose local moves associated with the differentiable part of the target density with (ii) a proximal step based on the non-differentiable part of the target density which provides sparse solutions such that small components are shrunk toward zero. Several implementations of the proximal step will be investigated adapted to different sparsity priors or allowing to perform variable selections, in high-dimensional settings. The performance of these new procedures are illustrated on both simulated and real data sets. Preliminary convergence results will also be presented.