🤖 AI Summary
This work addresses the challenge of efficient sampling and uncertainty quantification for posterior distributions under general constraints by proposing an extended weighted Bayesian bootstrap method, which is the first to generalize this approach to arbitrary constraint settings. By integrating convex optimization techniques, the method approximates the constrained posterior distribution even in scenarios where conventional approaches yield only point estimates. Theoretical analysis demonstrates that the asymptotic covariance of the generated samples aligns with that of the constrained maximum likelihood estimator, ensuring both computational efficiency and statistical validity. Empirical evaluations confirm the method’s broad applicability across diverse constrained Bayesian inference problems, including a successful application to uncertainty quantification for European option price surfaces.
📝 Abstract
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.