Bayesian Modular and Multiscale Regression

TL;DR

This paper introduces a Bayesian modular and multiscale regression method that decomposes high-dimensional spatial or temporal data into interpretable multiscale components, providing uncertainty quantification and efficient computation.

Contribution

The paper presents a novel Bayesian multiscale regression approach with a modular structure, enabling easy interpretation and uncertainty quantification for high-dimensional data.

Findings

Effective multiscale decomposition of complex data

Provides uncertainty quantification at different resolutions

Demonstrated successful application to brain image classification

Abstract

We tackle the problem of multiscale regression for predictors that are spatially or temporally indexed, or with a pre-specified multiscale structure, with a Bayesian modular approach. The regression function at the finest scale is expressed as an additive expansion of coarse to fine step functions. Our Modular and Multiscale (M&M) methodology provides multiscale decomposition of high-dimensional data arising from very fine measurements. Unlike more complex methods for functional predictors, our approach provides easy interpretation of the results. Additionally, it provides a quantification of uncertainty on the data resolution, solving a common problem researchers encounter with simple models on down-sampled data. We show that our modular and multiscale posterior has an empirical Bayes interpretation, with a simple limiting distribution in large samples. An efficient sampling algorithm is developed for posterior computation, and the methods are illustrated through simulation studies and an application to brain image classification. Source code is available as an R package at https://github.com/mkln/bmms.