论文标题
适当降低完全贝叶斯反转的后验分布
Appropriate reduction of the posterior distribution in fully Bayesian inversions
论文作者
论文摘要
贝叶斯反演从观察方程式和先前的信息中产生模型参数的后验分布，均由超参数加权。还为完全贝叶斯反转中的超参数引入了先验，使我们能够通过关节后端概率地评估模型参数和超参数。但是，即使在线性问题中，也无法解决如何从关节后部提取有关模型参数的有用信息。这项研究对完全贝叶斯反演的关节后验的适当维度降低了理论探索。我们将概率降低的方式分类为以下三类，重点是关节后部的边缘化：（1）使用联合后验而无需边缘化，（2）使用模型参数的边缘后验和（3）使用超参数的边缘后部。首先，我们得出了表征这些类别的几个分析结果。一个是线性逆问题中各个类别的概率最大化估计量的半分析表示套件。对于大量数据和模型参数，发现类别（1）和（2）类别的模式估计器在渐近上相同。我们还证明了类别（2）和（3）delta功能的渐近分布将其集中在其概率峰上，这预测了模型参数的两个不同的最佳估计。其次，我们进行了合成测试，并找到适当的降低是通过Akaike's Bayesian信息标准（ABIC）的类别实现的。其他还原类别对于许多模型参数而言是不合适的，在许多模型参数中，模型参数的边际后部的概率浓度不再是表示中心极限定理...
Bayesian inversion generates a posterior distribution of model parameters from an observation equation and prior information both weighted by hyperparameters. The prior is also introduced for the hyperparameters in fully Bayesian inversions and enables us to evaluate both the model parameters and hyperparameters probabilistically by the joint posterior. However, even in a linear inverse problem, it is unsolved how we should extract useful information on the model parameters from the joint posterior. This study presents a theoretical exploration into the appropriate dimensionality reduction of the joint posterior in the fully Bayesian inversion. We classify the ways of probability reduction into the following three categories focused on the marginalisation of the joint posterior: (1) using the joint posterior without marginalisation, (2) using the marginal posterior of the model parameters and (3) using the marginal posterior of the hyperparameters. First, we derive several analytical results that characterise these categories. One is a suite of semianalytic representations of the probability maximisation estimators for respective categories in the linear inverse problem. The mode estimators of categories (1) and (2) are found asymptotically identical for a large number of data and model parameters. We also prove the asymptotic distributions of categories (2) and (3) delta-functionally concentrate on their probability peaks, which predicts two distinct optimal estimates of the model parameters. Second, we conduct a synthetic test and find an appropriate reduction is realised by category (3), typified by Akaike's Bayesian information criterion (ABIC). The other reduction categories are shown inappropriate for the case of many model parameters, where the probability concentration of the marginal posterior of the model parameters is found no longer to mean the central limit theorem...
