Keywords: [ Bayesian Methods ] [ Bayesian Nonparametrics ] [ Probabilistic Inference - Models and Probabilistic Programming ]
There has been a growing realization of the potential of Bayesian machine learning as a platform that can provide both flexible modeling, accurate predictions as well as coherent uncertainty statements. In particular, Bayesian Additive Regression Trees (BART) have emerged as one of today’s most effective general approaches to predictive modeling under minimal assumptions. Statistical theoretical developments for machine learning have been mostly concerned with approximability or rates of estimation when recovering infinite dimensional objects (curves or densities). Despite the impressive array of available theoretical results, the literature has been largely silent about uncertainty quantification. In this work, we continue the theoretical investigation of BART initiated recently by Rockova and van der Pas (2017). We focus on statistical inference questions. In particular, we study the Bernstein-von Mises (BvM) phenomenon (i.e. asymptotic normality) for smooth linear functionals of the regression surface within the framework of non-parametric regression with fixed covariates. Our semi-parametric BvM results show that, beyond rate-optimal estimation, BART can be also used for valid statistical inference.