Sampling from the joint posterior distribution of Gaussian mixture models (GMMs) via standard Markov chain Monte Carlo (MCMC) imposes several computational challenges, which have prevented a broader full Bayesian implementation of these models. A growing body of literature has introduced the Weighted Likelihood Bootstrap and the Weighted Bayesian Bootstrap as alternatives to MCMC sampling. The core idea of these methods is to repeatedly compute maximum a posteriori (MAP) estimates on many randomly weighted posterior densities. These MAP estimates then can be treated as approximate posterior draws. Nonetheless, a central question remains unanswered: How to select the distribution of the random weights under arbitrary sample sizes. Thus, we introduce the Bayesian Optimized Bootstrap (BOB), a computational method to automatically select the weights distribution by minimizing, through Bayesian Optimization, a black-box and noisy version of the reverse KL divergence between the Bayesian posterior and an approximate posterior obtained via random weighting. Our proposed method allows for uncertainty quantification, approximate posterior sampling, and embraces recent developments in parallel computing. We show that BOB outperforms competing approaches in recovering the Bayesian posterior, while retaining key theoretical properties from existing methods. BOB's performance is demonstrated through extensive simulations, along with real-world data analyses.

翻译：通过标准马尔可夫链蒙特卡洛方法从高斯混合模型的联合后验分布中采样存在若干计算挑战，这阻碍了这些模型在更广泛的全贝叶斯框架中的实现。近年来，加权似然自助法和加权贝叶斯自助法作为MCMC采样的替代方案被提出。这些方法的核心思想是在多个随机加权的后验密度上重复计算最大后验估计，这些最大后验估计可被视为近似的后验样本。然而，一个关键问题仍未解决：如何在任意样本量下选择随机权重的分布。为此，我们提出贝叶斯优化自助法——一种通过贝叶斯优化最小化贝叶斯后验与随机加权所得近似后验之间的黑箱含噪逆KL散度，从而自动选择权重分布的计算方法。该方法支持不确定性量化与近似后验采样，并兼容并行计算的最新进展。实验表明，BOB在恢复贝叶斯后验方面优于其他竞争方法，同时保留了现有方法的关键理论特性。通过大量仿真实验与真实数据分析验证了BOB的性能。