Title: Exact Bayesian variable selection and averaging for block-diagonal designs
Authors: David Rossell - Universitat Pompeu Fabra (Spain) [presenting]
Omiros Papaspiliopoulos - UPF (Spain)
Abstract: Variable selection is considered when the gram matrix $X'X$ is block-orthogonal, e.g. as in principal component regression, wavelet regression or certain structures with interaction terms. Conditional on the residual variance $\phi$ most posterior quantities of interest have closed-form, but integrating out $\phi$ to duly account for uncertainty has proven challenging as in principle it requires a sum over $2^p$ models, and led to a number of adhoc solutions in the literature. We solved this bottleneck with a fast expression to integrate phi exactly (e.g. $O(p)$ operations when $X'X$ is diagonal), avoiding MCMC or other costly iterative schemes. Coupled with an efficient model search and other tricks the framework delivers extremely exact computation for large $p$, as we show in our examples. It is hoped that the computational framework can serve as a basis for efficient approximations under general $X'X$.