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Title: Bayesian variable selection under misspecified errors Authors:  David Rossell - Universitat Pompeu Fabra (Spain) [presenting]
Abstract: A main challenge in high-dimensional variable selection is enforcing sparsity. Because of theoretical and computational considerations most research are based on linear regression with Normal errors, but in actual applications errors may not be Normal, which can have a particularly marked effect on Bayesian inference. We extend the usual Bayesian variable selection framework to consider more flexible errors that capture asymmetry and heavier-than-normal tails. The error structure is learnt from the data, so that the model automatically reduces to Normal errors when the flexibility is not needed. We show convenient properties (log-likelihood concavity, simple computation) that render the approach practical in high dimensions. Further, although the models are slightly non-regular we show that one can obtain asymptotic sparsity rates under model misspecification. We also shed some light on an important consequence of model misspecification on Bayesian variable selection, namely a potential for a marked drop in power to detect truly active coefficients. This is confirmed in our examples, where we also illustrate computational advantages of inferring the residual distribution from the data.