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Sparseness of the regression coefficient vector is often a desirableproperty, since, among other benefits, sparseness improves interpretability. Inpractice, many true regression coefficients might be negligibly small, butnon-zero, which we refer to as quasi-sparseness. Spike-and-slab priors asintroduced in (Chipman et al., 2001) can be tuned to ignore very smallregression coefficients, and, as a consequence provide a trade-off betweenprediction accuracy and interpretability. However, spike-and-slab priors withfull support lead to inconsistent Bayes factors, in the sense that the Bayesfactors of any two models are bounded in probability. This is clearly anundesirable property for Bayesian hypotheses testing, where we wish thatincreasing sample sizes lead to increasing Bayes factors favoring the truemodel. The moment matching priors as in (Johnson and Rossell, 2012) can resolvethis issue, but are unsuitable for the quasi-sparse setting due to their fullsupport outside the exact value 0. As a remedy, we suggest disjunct supportspike and slab priors, for which we prove consistent Bayes factors in thequasi-sparse setting, and show experimentally fast growing Bayes factorsfavoring the true model. Several experiments on simulated and real data confirmthe usefulness of our proposed method to identify models with high effect size,while leading to better control over false positives than hard-thresholding.

Disjunct Support Spike and Slab Priors for Variable Selection in Regression under Quasi-sparseness