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

Regression shrinkage and variable selection are important concepts in high-dimensional statistics that allow the inference of robust models from large data sets. Bayesian methods achieve this by subjecting the model parameters to a prior distribution whose mass is centred around zero. In particular, the lasso and elastic net linear regression models employ a double-exponential distribution in their prior, which results in some maximum-likelihood regression coefficients being identically zero. Because of their ability to simultaneously perform parameter estimation and variable selection, these models have become enormously popular. However, there has been limited success in moving beyond maximum-likelihood estimation and deriving estimates for the posterior distribution of regression coefficients, due to a need for computationally expensive Gibbs sampling approaches to evaluate analytically intractable partition function integrals. Here, through the use of the Fourier transform, these integrals are expressed 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 distribution 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, thus allowing for Bayesian inference of much higher dimensional models than previously possible.