2020/10/08 by Sébastien Gadat, Gadat, Sébastien, Fabien Panloup +3
Computer Science · Mathematics · #60J45 #62C10 #65C05 #65C40 #93E35 #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (stat.ML) #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Statistical Methods and Inference #Statistics Theory (math.ST) #math.PR #math.ST #msc:60J45 #msc:62C10 #msc:65C05 #msc:65C40 #msc:93E35 #stat.ML #stat.TH
paper · pdf · doi:10.48550/arxiv.2010.06420
openalex publication_date 2020/10/08 · arxiv created 2022/02/14 · arxiv updated 2022/02/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/02
In this paper, we investigate the problem of computing Bayesian estimators using Langevin Monte-Carlo type approximation. The novelty of this paper is to consider together the statistical and numerical counterparts (in a general log-concave setting). More precisely, we address the following question: given n observations in ℝq distributed under an unknown probability ℙθ^⋆ with θ^⋆ ∈ ℝd , what is the optimal numerical strategy and its cost for the approximation of θ^⋆ with the Bayesian posterior mean? To answer this question, we establish some quantitative statistical bounds related to the underlying Poincaré constant of the model and establish new results about the numerical approximation of Gibbs measures by Cesaro averages of Euler schemes of (over-damped) Langevin diffusions. These last results include in particular some quantitative controls in the weakly convex case based on new bounds on the solution of the related Poisson equation of the diffusion.