A Mixing Time Lower Bound for a Simplified Version of BART
简化版 BART 的混合时间下界AI 解读 ↓
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简化版 BART 的混合时间下界
- 研究问题
- BART 的后验 MCMC 常出现混合缓慢；本文研究简化模型中混合时间是否随样本量快速增长，以及这对计算稳定性的含义。
- 方法与思路
- 作者将树之和简化为单棵树，并限制 MCMC 提议的移动类型，推导混合时间下界；随后通过 BART 模拟考察样本量与混合的关系。
- 材料陈述的贡献
- 摘要称，简化版的混合时间下界随数据量呈指数增长，BART 模拟也显示混合随数据量变慢，并据此建议增加并行链数量。
- 阅读关注点
- 应核对下界依赖的数据与提议条件、简化链和完整 BART 的差异，以及模拟中的混合诊断和多链策略能否缓解结果波动。
- 阅读边界
- 这是 arXiv 预印本，同行评审或期刊接收未经核实；指数下界针对特定简化版，不能直接推广为完整 BART 的普遍定理。
原文摘要
Bayesian Additive Regression Trees (BART) is a popular Bayesian non-parametric regression algorithm. The posterior is a distribution over sums of decision trees, and predictions are made by averaging approximate samples from the posterior. The combination of strong predictive performance and the ability to provide uncertainty measures has led BART to be commonly used in the social sciences, biostatistics, and causal inference. BART uses Markov Chain Monte Carlo (MCMC) to obtain approximate posterior samples over a parameterized space of sums of trees, but it has often been observed that the chains are slow to mix. In this paper, we provide the first lower bound on the mixing time for a simplified version of BART in which we reduce the sum to a single tree and use a subset of the possible moves for the MCMC proposal distribution. Our lower bound for the mixing time grows exponentially with the number of data points. Inspired by this new connection between the mixing time and the number of data points, we perform rigorous simulations on BART. We show qualitatively that BART's mixing time increases with the number of data points. The slow mixing time of the simplified BART suggests a large variation between different runs of the simplified BART algorithm and a similar large variation is known for BART in the literature. This large variation could result in a lack of stability in the models, predictions, and posterior intervals obtained from the BART MCMC samples. Our lower bound and simulations suggest increasing the number of chains with the number of data points.