Performing inference in Bayesian models requires sampling algorithms to draw samples from the posterior. This becomes prohibitively expensive as the size of data sets increase. Constructing approximations to the posterior which are cheap to evaluate is a popular approach to circumvent this issue. This begs the question of what is an appropriate space to perform approximation of Bayesian posterior measures. This manuscript studies the application of Bayes Hilbert spaces to the posterior approximation problem. Bayes Hilbert spaces are studied in functional data analysis in the context where observed functions are probability density functions and their application to computational Bayesian problems is in its infancy. This manuscript shall outline Bayes Hilbert spaces and their connection to Bayesian computation, in particular novel connections between Bayes Hilbert spaces, Bayesian coreset algorithms and kernel-based distances.

翻译：在贝叶斯模型中执行推断需要采样算法从后验分布中抽取样本。随着数据集规模的增大，该方法的计算成本变得极其高昂。构建易于评估的后验近似是规避这一问题的常用方法。这引出了一个关键问题：什么是适合执行贝叶斯后验测度近似的空间？本文研究了贝叶斯希尔伯特空间在后验近似问题中的应用。贝叶斯希尔伯特空间已在函数数据分析领域得到研究，其应用场景为观测函数是概率密度函数的情形，而在计算贝叶斯问题中的应用尚处于起步阶段。本文将阐述贝叶斯希尔伯特空间及其与贝叶斯计算的关联，特别是贝叶斯希尔伯特空间、贝叶斯核心集算法和基于核的距离之间新颖的关联性。