Bayesian Inference Using the Proximal Mapping: Uncertainty Quantification Under Varying Dimensionality

In statistical applications, it is common to encounter parameters supported on a varying or unknown dimensional space. Examples include the fused lasso regression, the matrix recovery under an unknown low rank, etc. Despite the ease of obtaining a point estimate via optimization, it is much more challenging to quantify their uncertainty. In the Bayesian framework, a major difficulty is that if assigning the prior associated with a p-dimensional measure, then there is zero posterior probability on any lower-dimensional subset with dimension d < p. To avoid this caveat, one needs to choose another dimension-selection prior on d, which often involves a highly combinatorial problem. To significantly reduce the modeling burden, we propose a new generative process for the prior: starting from a continuous random variable such as multivariate Gaussian, we transform it into a varying-dimensional space using the proximal mapping. This leads to a large class of new Bayesian models that can directly exploit the popular frequentist regularizations and their algorithms, such as the nuclear norm penalty and the alternating direction method of multipliers, while providing a principled and probabilistic uncertainty estimation. We show that this framework is well justified in the geometric measure theory, and enjoys a convenient posterior computation via the standard Hamiltonian Monte Carlo. We demonstrate its use in the analysis of the dynamic flow network data. Supplementary materials for this article are available online.

在统计学应用中，经常会遇到支撑集位于可变维度或未知维度空间的参数。举例而言，融合套索（fused lasso）回归、未知低秩约束下的矩阵恢复等均属于此类场景。尽管通过优化即可轻松得到点估计，但对这类参数的不确定性进行量化则要困难得多。在贝叶斯框架下，一个核心难题在于：若为参数指定与p维测度关联的先验分布，则任意维度d < p的低维子集的后验概率均为0。为规避这一局限，需针对维度d另行选择维度选择先验，而这往往会涉及高度复杂的组合优化问题。为大幅降低建模成本，本文提出一种针对先验的全新生成过程：从多元高斯等连续随机变量出发，通过邻近映射（proximal mapping）将其转换至可变维度空间。该方法可衍生出一大类全新的贝叶斯模型，能够直接复用主流的频率学派正则化方法及其配套算法（如核范数惩罚、交替方向乘子法），同时提供具备严格理论依据的概率化不确定性量化结果。本文证明该框架在几何测度论中具备充分的理论合理性，且可通过标准哈密顿蒙特卡洛（Hamiltonian Monte Carlo）算法实现便捷的后验推断。我们将该框架应用于动态流网络数据分析场景，并验证其有效性。本文的补充材料可在线获取。