BOB：用于高斯混合模型不确定性量化的贝叶斯优化自助法
BOB: Bayesian Optimized Bootstrap for Uncertainty Quantification in Gaussian Mixture Models
AI总结：
针对高斯混合模型中标准MCMC后验采样计算难度大、加权自助法随机权重难选的问题，提出贝叶斯优化自助法BOB，通过贝叶斯优化自动选权以最小化反向KL散度，实现更优的后验还原与不确定性量化。
摘要：
量化高斯混合模型（GMM）中不确定性的一种自然方式是借助贝叶斯方法。尽管如此，通过标准马尔可夫链蒙特卡洛（MCMC）从GMM的联合后验分布中采样存在若干计算挑战，阻碍了这些模型更广泛的完全贝叶斯实现。越来越多的文献提出了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。这类方法的核心思想是在大量随机加权的后验密度上反复计算最大后验（MAP）估计，随后可将这些MAP估计视为近似后验抽样。然而，一个核心问题仍未得到解答：如何在任意样本量下选择随机权重。因此，我们提出了贝叶斯优化自助法（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得到的近似后验之间的反向Kullback-Leibler（KL）散度的黑盒噪声版本，从而自动选择这些随机权重的计算方法。我们提出的方法在还原贝叶斯后验方面优于现有竞争方法，能提供更优的不确定性量化效果，同时保留了现有方法的关键渐近性质。我们通过大量模拟以及真实数据分析验证了BOB的性能。
英文摘要：
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.
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