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Decision Trees are some of the most popular machine learning models today dueto their out-of-the-box performance and interpretability. Often, Decision Treesmodels are constructed greedily in a top-down fashion via heuristic searchcriteria, such as Gini impurity or entropy. However, trees constructed in thismanner are sensitive to minor fluctuations in training data and are prone tooverfitting. In contrast, Bayesian approaches to tree construction formulatethe selection process as a posterior inference problem; such approaches aremore stable and provide greater theoretical guarantees. However, generatingBayesian Decision Trees usually requires sampling from complex, multimodalposterior distributions. Current Markov Chain Monte Carlo-based approaches forsampling Bayesian Decision Trees are prone to mode collapse and long mixingtimes, which makes them impractical. In this paper, we propose a new criterionfor training Bayesian Decision Trees. Our criterion gives rise to BCART-PCFG,which can efficiently sample decision trees from a posterior distributionacross trees given the data and find the maximum a posteriori (MAP) tree.Learning the posterior and training the sampler can be done in time that ispolynomial in the dataset size. Once the posterior has been learned, trees canbe sampled efficiently (linearly in the number of nodes). At the core of ourmethod is a reduction of sampling the posterior to sampling a derivation from aprobabilistic context-free grammar. We find that trees sampled via BCART-PCFGperform comparable to or better than greedily-constructed Decision Trees inclassification accuracy on several datasets. Additionally, the trees sampledvia BCART-PCFG are significantly smaller -- sometimes by as much as 20x.

Bayesian Decision Trees via Tractable Priors and Probabilistic Context-Free Grammars