The success of Bayesian inference with MCMC depends critically on Markov chains rapidly reaching the posterior distribution. Despite the plentitude of inferential theory for posteriors in Bayesian non-parametrics, convergence properties of MCMC algorithms that simulate from such ideal inferential targets are not thoroughly understood. This work focuses on the Bayesian CART algorithm which forms a building block of Bayesian Additive Regression Trees (BART). We derive upper bounds on mixing times for typical posteriors under various proposal distributions. Exploiting the wavelet representation of trees, we provide sufficient conditions for Bayesian CART to mix well (polynomially) under certain hierarchical connectivity restrictions on the signal. We also derive a negative result showing that Bayesian CART (based on simple grow and prune steps) cannot reach deep isolated signals in faster than exponential mixing time. To remediate myopic tree exploration, we propose Twiggy Bayesian CART which attaches/detaches entire twigs (not just single nodes) in the proposal distribution. We show polynomial mixing of Twiggy Bayesian CART without assuming that the signal is connected on a tree. Going further, we show that informed variants achieve even faster mixing. A thorough simulation study highlights discrepancies between spike-and-slab priors and Bayesian CART under a variety of proposals.

翻译：基于MCMC的贝叶斯推断成功的关键在于马尔可夫链能否快速收敛到后验分布。尽管贝叶斯非参数方法中后验分布的推断理论已十分丰富，但从这些理想推断目标中抽样的MCMC算法的收敛特性尚未被充分理解。本文聚焦于贝叶斯分类与回归树（Bayesian CART）算法——该算法是贝叶斯加性回归树（BART）的基础构建模块。我们推导了典型后验分布在多种提议分布下的混合时间上限。利用树的子波表示，我们给出了贝叶斯CART在信号满足特定层次连通性约束时实现良好（多项式阶）混合的充分条件。此外，我们得到一项负面结果：基于简单生长-剪枝步骤的贝叶斯CART在面对深层孤立信号时，其混合时间必然以指数级增长。为缓解树搜索的短视性，我们提出"细枝贝叶斯CART"（Twiggy Bayesian CART），其在提议分布中支持连接/分离整个细枝（而非单个节点）。我们证明：在不假设信号在树上连通的情况下，细枝贝叶斯CART仍可实现多项式阶混合。更进一步，我们展示了信息增强变体可实现更快的混合速度。全面的仿真研究揭示了在不同提议分布下，尖峰-板状先验与贝叶斯CART之间的显著差异。