We consider the problem of sampling from the posterior distribution of a $d$-dimensional coefficient vector $\boldsymbol{\theta}$, given linear observations $\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\theta}+\boldsymbol{\varepsilon}$. In general, such posteriors are multimodal, and therefore challenging to sample from. This observation has prompted the exploration of various heuristics that aim at approximating the posterior distribution. In this paper, we study a different approach based on decomposing the posterior distribution into a log-concave mixture of simple product measures. This decomposition allows us to reduce sampling from a multimodal distribution of interest to sampling from a log-concave one, which is tractable and has been investigated in detail. We prove that, under mild conditions on the prior, for random designs, such measure decomposition is generally feasible when the number of samples per parameter $n/d$ exceeds a constant threshold. We thus obtain a provably efficient (polynomial time) sampling algorithm in a regime where this was previously not known. Numerical simulations confirm that the algorithm is practical, and reveal that it has attractive statistical properties compared to state-of-the-art methods.

翻译：我们考虑从$d$维系数向量$\boldsymbol{\theta}$的后验分布中采样的问题，其中给定线性观测$\boldsymbol{y} = \boldsymbol{X}\boldsymbol{\theta}+\boldsymbol{\varepsilon}$。一般而言，此类后验分布具有多峰特性，因此难以采样。这一现象促使学界探索了多种旨在逼近后验分布的启发式方法。本文研究一种基于将后验分布分解为对数凹的简单乘积测度混合体的新方法。该分解使我们能够将目标多峰分布的采样问题转化为对数凹分布的采样问题，后者具有可处理性且已被深入研究。我们证明，在随机设计条件下，当单位参数样本量$n/d$超过常数阈值时，只要先验分布满足温和条件，此类测度分解通常可行。由此我们在以往未知的可行区域内获得了一种可证明高效（多项式时间）的采样算法。数值模拟证实该算法具有实用性，并揭示其相较于前沿方法具有更优的统计特性。