Title: Analytic solution and stationary phase approximation for the 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.