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The lasso and elastic net linear regression models impose adouble-exponential prior distribution on the model parameters to achieveregression shrinkage and variable selection, allowing the inference of robustmodels from large data sets. However, there has been limited success inderiving estimates for the full posterior distribution of regressioncoefficients in these models, due to a need to evaluate analyticallyintractable partition function integrals. Here, the Fourier transform is usedto express these integrals as complex-valued oscillatory integrals over"regression frequencies". This results in an analytic expansion and stationaryphase approximation for the partition functions of the Bayesian lasso andelastic net, where the non-differentiability of the double-exponential priorhas so far eluded such an approach. Use of this approximation leads to highlyaccurate numerical estimates for the expectation values and marginal posteriordistributions of the regression coefficients, and allows for Bayesian inferenceof much higher dimensional models than previously possible.

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