Bayesian Deep Net GLM and GLMM

TL;DR

This paper introduces a Bayesian deep neural network framework integrated with generalized linear models and mixed models, offering high prediction accuracy and uncertainty quantification, with efficient computational methods for high-dimensional inference.

Contribution

It develops a novel Bayesian deep learning approach for GLMs and GLMMs using Gaussian variational approximation and natural gradient optimization, enabling flexible modeling with variable selection.

Findings

High prediction accuracy demonstrated on simulated and real data

Effective uncertainty quantification in predictions

Favorable comparison with Bayesian additive regression trees (BART)

Abstract

Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. The consideration of neural networks with random effects is not widely used in the literature, perhaps because of the computational challenges of incorporating subject specific parameters into already complex models. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parametrization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and the results compare favourably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab, R and Python implementing the proposed methods are available at https://github.com/VBayesLab