A natural way to quantify uncertainties in Gaussian mixture models (GMMs) is through Bayesian methods. That said, sampling from the joint posterior distribution of 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 random weights under arbitrary sample sizes. We, therefore, introduce the Bayesian Optimized Bootstrap (BOB), a computational method to automatically select these random weights by minimizing, through Bayesian Optimization, a black-box and noisy version of the reverse Kullback-Leibler (KL) divergence between the Bayesian posterior and an approximate posterior obtained via random weighting. Our proposed method outperforms competing approaches in recovering the Bayesian posterior, it provides a better uncertainty quantification, and it retains key asymptotic properties from existing methods. BOB's performance is demonstrated through extensive simulations, along with real-world data analyses.

翻译：量化高斯混合模型（GMMs）不确定性的自然途径是采用贝叶斯方法。然而，通过标准马尔可夫链蒙特卡洛（MCMC）从GMMs的联合后验分布中采样存在多项计算难题，这阻碍了这些模型更广泛的完全贝叶斯实现。日益增多的文献引入了加权似然Bootstrap和加权贝叶斯Bootstrap作为MCMC采样的替代方案。这些方法的核心思想是在多个随机加权的后验密度上反复计算最大后验（MAP）估计，这些MAP估计可被视为近似后验抽取。尽管如此，一个核心问题仍未解决：如何在任意样本量下选择随机权重。为此，我们提出了贝叶斯优化引导的Bootstrap（BOB），这是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反向KL散度的黑箱含噪版本，来自动选择这些随机权重的计算方法。我们的方法在恢复贝叶斯后验方面优于现有竞争方法，提供了更优的不确定性量化，并保留了现有方法的关键渐近性质。通过大量仿真实验和真实数据分析验证了BOB的性能。