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Abstract:
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The widespread popularity of Bayesian tree-structured regression methods has raised considerable interest in theoretical understanding of their empirical success. This paper affords new insights about Bayesian CART in the context of structured wavelet shrinkage under the white noise model. We exhibit precise connections between tree-shaped sparsity priors and unstructured spike-and-slab priors, which are regarded as ideal but are rather theoretical in nature. We show that the more practical Bayesian CART priors lead to adaptive rate-minimax posterior concentration in the l? sense, performing nearly as well as the theoretical ideal (up to a log term). To further explore the benefits of structured shrinkage, we propose the g-prior for trees, which departs from the typical wavelet product priors by harnessing correlation induced by the tree topology. While the majority of wavelet theoretical results for CART focus on dyadic trees, here we do not require that splits are at dyadic locations. Building on the l? adaptation property, we provide new fully non-parametric and adaptive Bernstein-von Mises statements for Bayesian CART using multiscale techniques.
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