Tree-based priors for probability distributions are usually specified using a predetermined, data-independent collection of candidate recursive partitions of the sample space. To characterize an unknown target density in detail over the entire sample space, candidate partitions must have the capacity to expand deeply into all areas of the sample space with potential non-zero sampling probability. Such an expansive system of partitions often incurs prohibitive computational costs and makes inference prone to overfitting, especially in regions with little probability mass. Thus, existing models typically make a compromise and rely on relatively shallow trees. This hampers one of the most desirable features of trees, their ability to characterize local features, and results in reduced statistical efficiency. Traditional wisdom suggests that this compromise is inevitable to ensure coherent likelihood-based reasoning in Bayesian inference, as a data-dependent partition system that allows deeper expansion only in regions with more observations would induce double dipping of the data. We propose a simple strategy to restore coherency while allowing the candidate partitions to be data-dependent, using Cox's partial likelihood. Our partial likelihood approach is broadly applicable to existing likelihood-based methods and, in particular, to Bayesian inference on tree-based models. We give examples in density estimation in which the partial likelihood is endowed with existing priors on tree-based models and compare with the standard, full-likelihood approach. The results show substantial gains in estimation accuracy and computational efficiency from adopting the partial likelihood.

翻译：基于树结构的概率分布先验通常通过一组预先确定且独立于数据的候选递归分割来定义。为了在整个样本空间中详细刻画未知目标密度，候选分割必须能够深入扩展至样本空间所有可能具有非零采样概率的区域。这种广泛的分割系统往往导致难以承受的计算成本，并使得推断容易过拟合，特别是在概率质量较低的区域。因此，现有模型通常做出妥协，依赖相对浅层的树结构。这削弱了树模型最具吸引力的特性之一——刻画局部特征的能力，并导致统计效率降低。传统观点认为，这种妥协对于确保贝叶斯推断中基于似然的连贯推理是不可避免的，因为允许仅在观测较多的区域进行更深扩展的数据依赖分割系统会导致数据的重复利用。我们提出一种简单策略，通过使用Cox的部分似然，在允许候选分割依赖数据的同时恢复推理的连贯性。我们的部分似然方法广泛适用于现有的基于似然的方法，特别是树模型的贝叶斯推断。我们以密度估计为例，将部分似然与现有树模型先验结合，并与标准的全似然方法进行比较。结果表明，采用部分似然在估计精度和计算效率方面均有显著提升。