Analytic solution and stationary phase approximation for the Bayesian lasso and elastic net

TL;DR

This paper introduces an analytic and stationary phase approximation method for the Bayesian lasso and elastic net models, enabling efficient computation of posterior distributions in high-dimensional settings.

Contribution

It develops a novel Fourier transform-based approach to evaluate intractable partition functions, overcoming non-differentiability issues of the double-exponential prior.

Findings

Provides highly accurate estimates of posterior expectations.

Enables Bayesian inference in higher-dimensional models.

Offers a new analytical framework for Bayesian lasso and elastic net.

Abstract

The lasso and elastic net linear regression models impose a double-exponential prior distribution on the model parameters to achieve regression shrinkage and variable selection, allowing the inference of robust models from large data sets. However, there has been limited success in deriving estimates for the full posterior distribution of regression coefficients in these models, due to a need to evaluate analytically intractable partition function integrals. Here, the Fourier transform is used to express these integrals as complex-valued oscillatory integrals over "regression frequencies". This results in an analytic expansion and stationary phase approximation for the partition functions of the Bayesian lasso and elastic net, where the non-differentiability of the double-exponential prior has so far eluded such an approach. Use of this approximation leads to highly accurate numerical estimates for the expectation values and marginal posterior distributions of the regression coefficients, and allows for Bayesian inference of much higher dimensional models than previously possible.