UCSC-SOE-21-02: Convergence Rate for Predictive Densities of Bayes Generalized Linear Models with a Scalar Response and Symm. Tensor Predictor

R. Guhaniyogi and S. Guha
05/12/2021 01:14 PM
This article investigates statistical convergence rates for predictive densities of a novel Bayesian generalized linear model (GLM) framework with a scalar response and a symmetric tensor predictor with labeled "nodes." GLM frameworks involving a symmetric tensor predictor and a scalar response may appear in a variety of real life applications, including diffusion weighted magnetic resonance imaging (DWI) and functional magnetic resonance imaging (fMRI), among others. This article speci fically focuses on a class of such models where
the over-arching goal is to identify nodes and cells of the symmetric tensor influential in predicting the response. We establish a near optimal convergence rate for the posterior predictive density from the proposed model to the true density, depending on how the number of tensor nodes grows with the sample size. Moreover, we show that the method has adaptivity to the unknown rank of the true tensor, i.e., the near optimal rate is achieved even if the rank of the true tensor coefficient is not known a priori.