Estimation of parameters that obey specific constraints is crucial in statistics and machine learning; for example, when parameters are required to satisfy boundedness, monotonicity, or linear inequalities. Traditional approaches impose these constraints via constraint-specific transformations or sampling approaches, or by truncating the posterior distribution. Such methods often result in computational challenges, limited flexibility, and a lack of generality. We propose a generalized framework for constrained Bayesian inference by projecting the unconstrained posterior distribution into the space of the parameter constraints, providing a computationally efficient and easily implementable solution for a large class of problems. We rigorously establish the theoretical foundations of the projected posterior distribution, as well as providing asymptotic results for posterior consistency, posterior contraction, and optimal coverage properties. Our methodology is validated through both theoretical arguments and practical applications.

翻译：在统计学与机器学习中，对满足特定约束的参数进行估计至关重要，例如参数需满足有界性、单调性或线性不等式等条件。传统方法通过约束特定的变换、采样方法或截断后验分布来施加这些约束，但此类方法往往面临计算困难、灵活性有限且缺乏普适性等问题。本文提出一种广义约束贝叶斯推断框架，通过将无约束后验分布投影到参数约束空间中，为一大类问题提供了计算高效且易于实现的解决方案。我们严格建立了投影后验分布的理论基础，并给出了后验一致性、后验收缩及最优覆盖性质的新近结果。该方法的有效性通过理论论证和实际应用得到了双重验证。