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In the field of decision trees, most previous studies have difficultyensuring the statistical optimality of a prediction of new data and suffer fromoverfitting because trees are usually used only to represent predictionfunctions to be constructed from given data. In contrast, some studies,including this paper, used the trees to represent stochastic data observationprocesses behind given data. Moreover, they derived the statistically optimalprediction, which is robust against overfitting, based on the Bayesian decisiontheory by assuming a prior distribution for the trees. However, these studiesstill have a problem in computing this Bayes optimal prediction because itinvolves an infeasible summation for all division patterns of a feature space,which is represented by the trees and some parameters. In particular, an openproblem is a summation with respect to combinations of division axes, i.e., theassignment of features to inner nodes of the tree. We solve this by a Markovchain Monte Carlo method, whose step size is adaptively tuned according to aposterior distribution for the trees.

Prediction Algorithms Achieving Bayesian Decision Theoretical Optimality Based on Decision Trees as Data Observation Processes