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

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.Comment: Switched to new NeurIPS style file; 11 pages, 3 figures + appendices 29 pages, 3 supplementary figure

Priors on the Variance in Sparse Bayesian Learning; the demi-Bayesian Lasso

We explore the use of proper priors for variance parameters of certain sparse Bayesian regression models. This leads to a connection between sparse Bayesian learning (SBL) models (Tipping, 2001) and the recently proposed Bayesian Lasso (Park and Casella, 2008). We outline simple modifications of existing algorithms to solve this new variant which essentially uses type-II maximum likelihood to fit the Bayesian Lasso model. We also propose an Elastic-net (Zou and Hastie, 2005) heuristic to help with modeling correlated inputs. Experimental results show the proposals to compare favorably to both the Lasso and traditional and more recent sparse Bayesian algorithms

Regularized brain reading with shrinkage and smoothing

Functional neuroimaging measures how the brain responds to complex stimuli. However, sample sizes are modest, noise is substantial, and stimuli are high dimensional. Hence, direct estimates are inherently imprecise and call for regularization. We compare a suite of approaches which regularize via shrinkage: ridge regression, the elastic net (a generalization of ridge regression and the lasso), and a hierarchical Bayesian model based on small area estimation (SAE). We contrast regularization with spatial smoothing and combinations of smoothing and shrinkage. All methods are tested on functional magnetic resonance imaging (fMRI) data from multiple subjects participating in two different experiments related to reading, for both predicting neural response to stimuli and decoding stimuli from responses. Interestingly, when the regularization parameters are chosen by cross-validation independently for every voxel, low/high regularization is chosen in voxels where the classification accuracy is high/low, indicating that the regularization intensity is a good tool for identification of relevant voxels for the cognitive task. Surprisingly, all the regularization methods work about equally well, suggesting that beating basic smoothing and shrinkage will take not only clever methods, but also careful modeling.Comment: Published at http://dx.doi.org/10.1214/15-AOAS837 in the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org