Augmentation Samplers for Multinomial Probit Bayesian Additive Regression Trees

The multinomial probit (MNP) framework is based on a multivariate Gaussian latent structure, allowing for natural extensions to multilevel modeling. Unlike multinomial logistic models, MNP does not assume independent alternatives. Kindo, Wang, and Peña proposed multinomial probit BART (MPBART) to accommodate Bayesian additive regression trees (BART) formulation in MNP. The posterior sampling algorithms for MNP and MPBART are collapsed Gibbs samplers. Because the collapsing augmentation strategy yields a geometric rate of convergence no greater than that of a standard Gibbs sampling step, it is recommended whenever computationally feasible (Liu; Imai and van Dyk). While this strategy necessitates simple sampling steps and a reasonably fast converging Markov chain, the complexity of the stochastic search for posterior trees may undermine its benefit. We address this problem by sampling posterior trees conditional on the constrained parameter space and compare our proposals to that of Kindo, Wang, and Peña, who sample posterior trees based on an augmented parameter space. In terms of MCMC convergence and posterior predictive accuracy, our proposals outperform the augmented tree sampling approach. We also show that the theoretical mixing rates of our proposals are guaranteed to be no greater than the augmented tree sampling approach. Appendices and codes for simulations and demonstrations are available online.

多项Probit（multinomial probit, MNP）框架基于多元高斯潜在结构，可自然拓展至多层建模场景。与多项Logistic模型不同，MNP无需假设备选方案相互独立。Kindo、Wang与Peña提出了多项Probit贝叶斯可加回归树（multinomial probit BART, MPBART），以将贝叶斯可加回归树（Bayesian additive regression trees, BART）的建模范式融入MNP框架。MNP与MPBART的后验抽样算法均采用折叠吉布斯采样器。由于折叠增强策略的几何收敛速率不高于标准吉布斯采样步的收敛速率，因此在计算资源可行时推荐使用该策略（Liu；Imai与van Dyk）。尽管该策略仅需简单抽样步骤且可使马尔可夫链实现较快收敛，但针对后验树的随机搜索复杂度可能削弱其优势。为此，我们提出基于约束参数空间的后验树抽样方法，并与Kindo、Wang与Peña基于增强参数空间的后验树抽样方案展开对比。在马尔可夫链蒙特卡洛（MCMC）收敛性与后验预测精度方面，我们提出的方案优于增强树抽样方法。我们还证实，所提方案的理论混合速率保证不高于增强树抽样方法。本研究的仿真与演示所用附录及代码均可在线获取。