Abstract

The use of nondifferentiable priors in Bayesian statistics has become increasingly popular, in particular in Bayesian imaging analysis. Current state-of-the-art methods are approximate in the sense that they replace the posterior with a smooth approximation via Moreau-Yosida envelopes, and apply gradient-based discretized diffusions to sample from the resulting distribution. We characterize the error of the Moreau-Yosida approximation and propose a novel implementation using underdamped Langevin dynamics. In misson-critical cases, however, replacing the posterior with an approximation may not be a viable option. Instead, we show that piecewise-deterministic Markov processes (PDMP) can be used for exact posterior inference from distributions satisfying almost everywhere differentiability. Furthermore, in contrast with diffusion-based methods, the suggested PDMP-based samplers place no assumptions on the prior shape, nor require access to a computationally cheap proximal operator, and consequently have a much broader scope of application. Through detailed numerical examples, including a nondifferentiable circular distribution and a nonconvex genomics model, we elucidate the relative strengths of these sampling methods on problems of moderate to high dimensions, underlining the benefits of PDMP-based methods when accurate sampling is decisive. Supplementary materials for this article are available online.

摘要

在贝叶斯统计中使用不可微先验变得越来越流行，特别是在贝叶斯成像分析中。当前最先进的方法在某种意义上是近似的，因为它们通过 Moreau-Yosida 包络用平滑近似代替后验，并将基于梯度的离散化扩散应用于从所得分布中采样。我们描述了 Moreau-Yosida 近似的误差，并提出了一种使用欠阻尼 Langevin 动力学的新实现。然而，在错误关键的情况下，用近似值替换后验可能不是一个可行的选择。相反，我们表明分段确定性马尔可夫过程 (PDMP) 可用于从满足几乎处处可微性的分布进行精确的后验推理。此外，与基于扩散的方法相比，建议的基于 PDMP 的采样器不对先验形状进行假设，也不需要访问计算成本低的近端算子，因此具有更广泛的应用范围。通过详细的数值示例，包括不可微循环分布和非凸基因组学模型，我们阐明了这些采样方法在中高维问题上的相对优势，强调了在准确采样具有决定性作用时基于 PDMP 的方法的优势。本文的补充材料可在线获取。包括不可微循环分布和非凸基因组学模型，我们阐明了这些采样方法在中高维问题上的相对优势，强调了在准确采样具有决定性作用时基于 PDMP 的方法的优势。本文的补充材料可在线获取。包括不可微循环分布和非凸基因组学模型，我们阐明了这些采样方法在中高维问题上的相对优势，强调了在准确采样具有决定性作用时基于 PDMP 的方法的优势。本文的补充材料可在线获取。