On Semi-parametric Bernstein-von Mises Theorems for BART
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Few methods in Bayesian non-parametric statistics/ machine learning have received as much attention as Bayesian Additive Regression Trees (BART). While BART is now routinely performed for prediction tasks, its theoretical properties began to be understood only very recently. In this work, we continue the theoretical investigation of BART initiated by Rockova and van der Pas (2017). 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. As with other adaptive priors, the BvM phenomenon may fail when the regularities of the functional and the truth are not compatible. To overcome the curse of adaptivity under hierarchical priors, we induce a self-similarity assumption to ensure convergence towards a single Gaussian distribution as opposed to a Gaussian mixture. Similar qualitative restrictions on the functional parameter are known to be necessary for adaptive inference. Many machine learning methods lack coherent probabilistic mechanisms for gauging uncertainty. BART readily provides such quantification via posterior credible sets. The BvM theorem implies that the credible sets are also confidence regions with the same asymptotic coverage. This paper presents the first asymptotic normality result for BART priors, providing another piece of evidence that BART is a valid tool from a frequentist point of view.
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