UCSC-SOE-20-12: Convergence Rate for Predictive Densities of Bayesian Generalized Models with a Scalar Response and Symmetric Tensor Predictor

Rajarshi Guhaniyogi and Sharmistha Guha
10/26/2020 09:22 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." Such a framework may appear in a
variety of applications, including diffusion weighted magnetic
resonance imaging (DWI) and functional magnetic resonance
imaging (fMRI), among others. This article specifically focuses
on a class of models where the over-arching goal is to identify
influential nodes and cells of the symmetric tensor. 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.