A Unified Bayesian Inference Framework for Generalized Linear Models

TL;DR

This paper introduces a unified Bayesian inference framework for generalized linear models that simplifies complex problems into standard linear models, offering new insights and extensions for existing algorithms like AMP, VAMP, and SBL, with demonstrated effectiveness in compressed sensing.

Contribution

It proposes a novel unified Bayesian inference framework for GLMs, connecting and extending existing algorithms such as AMP, VAMP, and SBL.

Findings

GLM version of AMP is equivalent to GAMP.

Framework effectively applies to 1-bit quantized compressed sensing.

Provides new perspectives and extensions for established algorithms.

Abstract

In this letter, we present a unified Bayesian inference framework for generalized linear models (GLM) which iteratively reduces the GLM problem to a sequence of standard linear model (SLM) problems. This framework provides new perspectives on some established GLM algorithms derived from SLM ones and also suggests novel extensions for some other SLM algorithms. Specific instances elucidated under such framework are the GLM versions of approximate message passing (AMP), vector AMP (VAMP), and sparse Bayesian learning (SBL). It is proved that the resultant GLM version of AMP is equivalent to the well-known generalized approximate message passing (GAMP). Numerical results for 1-bit quantized compressed sensing (CS) demonstrate the effectiveness of this unified framework.