We prove the weighted Bayesian bootstrap, a method for approximate sampling of a posterior distribution, can be extended to sample from general constrained posterior distributions under mild assumptions. The method entails a simple algorithm that can take advantage of fast tools from convex optimization. Under regularity conditions, we show the asymptotic distribution of samples from the constrained weighted Bayesian bootstrap has a covariance matching the restricted maximum likelihood estimator, an efficient estimator. We assess the method empirically on a variety of constrained Bayesian problems, demonstrating broad applicability of the method as well as advantages over existing peer methods. The constrained weighted Bayesian bootstrap quickly samples from constrained posteriors, providing adequate uncertainty quantification for problems typically solved via optimization methods designed to deliver only a point estimate. As a case study, using constraints required in European-style option prices, uncertainty estimates of an option pricing surface are derived with constrained weighted Bayesian bootstrap.

翻译：我们证明，加权贝叶斯自助法（一种用于后验分布近似采样的方法）可以在温和假设下扩展为从一般受约束后验分布中采样。该方法采用简洁算法，能够利用凸优化中的快速工具。在正则性条件下，我们证明受约束加权贝叶斯自助法样本的渐近分布具有与受限极大似然估计量（一种有效估计量）相匹配的协方差。我们通过多种受约束贝叶斯问题对方法进行经验评估，展示了该方法的广泛适用性以及相较于现有同类方法的优势。受约束加权贝叶斯自助法能够快速从受约束后验中采样，为通常仅通过提供点估计的优化方法解决的问题提供了充分的不确定性量化。作为案例研究，我们利用欧式期权价格所需的约束条件，通过受约束加权贝叶斯自助法推导出了期权定价曲面的不确定性估计。