We present a simple method to obtain optimal posterior distributions and improve the quality of Bayesian inference with reduced human and computational effort. Bayes' Theorem is reformulated in the language of statistical mechanics, wherein an improved posterior -- referred to as a tempered posterior -- is defined analogously to a canonical probability distribution at temperature $τ$. Wang-Landau sampling is used to obtain the density of states of the posterior probability, and signals analogous to those of phase transitions are extracted from a single simulation. In addition, the transition temperature is easily identified, providing the tempered posterior with optimal predictive performance. We demonstrate the efficacy of the method on a real-world problem in materials science (equation of state modeling) with messy data, a high-dimensional and correlated input parameter space, and "frustration" among model outputs.

翻译：本文提出一种简洁方法，以获取最优后验分布并提升贝叶斯推断质量，同时降低人工与计算开销。我们将贝叶斯定理重新表述为统计力学语言，其中改进后的后验——称为温度缩放后验——被类比定义为温度τ下的正则概率分布。通过Wang-Landau采样获取后验概率的状态密度，并基于单次模拟提取类似相变特征信号。此外，相变温度可被精确识别，从而赋予温度缩放后验最优的预测性能。我们以材料科学中的实际问题（状态方程建模）验证该方法有效性：该问题包含噪声数据、高维关联参数空间以及模型输出间的“阻挫”现象。