S-values and Bayesian weighted all-subsets regressions

This paper compares and contrasts Bayesian variable-exclusion methods proposed by Eduardo Ley and coauthors with methods proposed by Raftery and Sala-i-Martin et al. and with the s-values proposed by myself. A distinction is drawn between estimation uncertainty which is the focus of Ley׳s research and model ambiguity which arises in Ley׳s work and is the focus of my own recent proposal. The discussion is organized around the prior covariance matrix, which needs to be diagonal to support all-subsets regressions. The basic question addressed here is: what aspects of the prior covariance matrix can be taken as known, what aspects can be estimated and what aspects require a sensitivity analysis because they are neither known nor estimable. When diagonality is in doubt, we are more-or-less forced into a model ambiguity sensitivity mode because the data are never rich enough credibly to estimate the full prior covariance matrix. When diagonality is assumed, the data evidence, though very limited, can help to estimate the diagonal elements, but this literature has not yet produced a compelling conventional treatment which will necessarily include both estimation uncertainty and model ambiguity as they relate both to the diagonal values and to the rest of the prior covariance matrix. But there has been a lot of progress.