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