Projected priors were originally introduced to accommodate parameter constraints, but have recently regained popularity due to their ability to assign probability mass to low-dimensional parameter sets, such as the spaces of sparse vectors, directed acyclic graphs, or transport plans. When employed as a transformation of random variables, projection is especially useful, since its contraction property not only preserves probability concentration, but also often preserves differentiability for gradient-based posterior computation. On the other hand, unless the projection can be obtained by some non-iterative algorithm, posterior computation can be expensive because it requires nesting an iterative optimization routine within each Markov chain Monte Carlo iteration. In this article, inspired by the success of continuous shrinkage models as replacements for discrete spike-and-slab priors, we propose a continuous relaxation of projected priors. The key idea is to quantify the duality gap between the primal projection loss and the dual objective, and impose a probabilistic prior that shrinks this gap toward zero. The resulting gap-shrinkage prior has a tractable form, does not require running an optimization subroutine inside each posterior update, and puts probability mass near the exact projection. We demonstrate useful properties of gap-shrinkage priors, including connections to global-local shrinkage priors, broad applicability to generalized projection functions, and competitive performance in posterior contraction. We apply the gap-shrinkage model to a marketing data analysis aimed at identifying important predictor effects on multivariate grocery-shopping decisions.

翻译：投影先验最初是为满足参数约束而提出的，但近年来因其能够将概率质量分配给低维参数集（如稀疏向量空间、有向无环图空间或传输计划空间）而重新受到关注。当用作随机变量变换时，投影尤为有用，因其收缩特性不仅能保持概率集中性，还能在基于梯度的后验计算中保持可微性。然而，除非投影可通过非迭代算法实现，否则后验计算可能代价高昂——因为需要在每次马尔可夫链蒙特卡洛迭代中嵌套迭代优化步骤。受连续收缩模型成功替代离散的spike-and-slab先验的启发，本文提出投影先验的一种连续松弛方法。核心思想是量化原始投影损失函数与对偶目标函数之间的对偶间隙，并通过施加概率先验使该间隙收缩至零。由此得到的间隙收缩先验具有可处理形式，无需在后验更新中运行优化子程序，且能将概率质量集中在精确投影附近。我们展示了间隙收缩先验的有用特性，包括与全局-局部收缩先验的联系、对广义投影函数的广泛适用性，以及在后验收缩中的竞争性能。将该模型应用于旨在识别多元购物决策中重要预测变量效应的市场数据分析。