论文标题
关于贝叶斯后期平均策略的成本
On the cost of Bayesian posterior mean strategy for log-concave models
论文作者
论文摘要
在本文中，我们研究了使用Langevin Monte-Carlo型近似计算贝叶斯估计器的问题。本文的新颖性是将统计和数值对应物（在一般的对数符合设置）一起考虑。更准确地说，我们解决以下问题：给定$ n $观察，以$ \ m \ m \ m \ m}^q $在未知的概率下分布$ \ mathbb {p} _ {θ^\ star} $，带有$θ^\ star \ in \ mathb {r}^d $的$ $ $ $ $ $ a $ oft $ oft $ oft $ oft $ ofter $ oft $ oft $，贝叶斯后部的意思？为了回答这个问题，我们建立了与模型的基本庞加莱常数相关的一些定量统计界，并通过（通过过度阻尼）langevin扩散的Euler方案的cesaro平均值建立了有关Gibbs测量的数值近似的新结果。这些最后的结果在弱凸情况下特别包括基于扩散相关泊松方程的新界限中的一些定量控制。
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 $\mathbb{R}^q$ distributed under an unknown probability $\mathbb{P}_{θ^\star}$ with $θ^\star \in \mathbb{R}^d$ , what is the optimal numerical strategy and its cost for the approximation of $θ^\star$ 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.
