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 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, including bounded-monotonic regression and emulation of a computer model with directional outputs.

翻译：参数估计在统计学与机器学习中至关重要，尤其是当参数需满足特定约束条件时，例如有界性、单调性或线性不等式。传统方法通常通过约束特定的变换或截断后验分布来施加这些约束，但此类方法常导致计算困难、灵活性受限且缺乏普适性。本文提出一种广义的约束贝叶斯推断框架，通过将无约束后验分布投影至参数约束空间，为一大类问题提供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渐近结果。通过理论论证与实际应用（包括有界单调回归和具有方向输出的计算机模型仿真），验证了所提方法的有效性。