In our upcoming Bayesian Working group meeting on Monday (05/05), 1:10 - 2 pm in Snedecor 2113, Dr. Pulong Ma will be presenting.

Title: On Posterior Concentration of Bayesian Gaussian Process Trees

Abstract:

Gaussian processes (GP) and trees are both flexible machine learning tools for nonparametric regression with a wide range of successful applications from agriculture to E-commerce. GP is well-known for spatial smoothing but lacks the ability to model heterogeneity and to scale well for large datasets while tree-based models are designed to capture local heterogeneity and can allow fast computations.  In recent years, there is a growing interest in developing hybrid models of GP and Bayesian trees to better capture heterogeneity and allow fast computations simultaneously, by exploiting their modeling advantages. While these models are appealing for statistical modeling and data analysis in modern complex applications, their theoretical properties have not been well-understood in part because theoretical analysis of Bayesian trees has progressed substantially only in the past few years. In nonparametric regression framework, it remains unclear on whether hybrid models of GP and Bayesian trees can provide theoretical guarantees. In this talk, I introduce a class of Bayesian GP tree models, where a stochastic tree prior is constructed by random recursive partitioning of the input domain; given such a partition tree, independent GPs are assumed over leaf nodes. Then I will examine the posterior concentration properties of the resulting Bayesian GP tree model under various settings. In particular, I show that the Bayesian GP tree model enjoys posterior consistency for both fixed covariates and random covariates as the sample size goes to infinity. I will also discuss the posterior contraction rate under sup-norm for estimating certain class of Hölderian functions.