Compatibility of Prior Specifications Across Linear Models

Abstract

Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links with related objective approaches. Given an encompassing, or full, model together with a prior on its parameter space, we review and summarize a few procedures for deriving priors under a submodel, namely marginalization, conditioning, and Kullback--Leibler projection. These techniques are illustrated and discussed with reference to variable selection in linear models adopting a conventional $g$-prior; comparisons with existing standard approaches are provided. Finally, the relative merits of each procedure are evaluated through simulated and real data sets.Comment: Published in at http://dx.doi.org/10.1214/08-STS258 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

Bayesian hypothesis testing for psychologists: A tutorial on the Savage–Dickey method

“…/ 0 ; this effectively means that H 1 reduces to H 0 -it is therefore reasonable to assume that pðwj/ ! / 0 ; H 1 Þ ¼ pðwjH 0 Þ (but see Consonni & Veronese, 2008). In other words, when / !…”

“…This prior specification is intuitively plausible, but it has two disadvantages. The first disadvantage is that it is implicitly assumed that the common nuisance parameters fulfill exactly the same roles, whether they are part of H 0 or H 1 ; some people believe this assumption is too strict (for a discussion see Consonni & Veronese, 2008). The second disadvantage is that the priors are constructed by conditioning on an event that has probability zero, namely / !…”

“…/ 0 . This way of conditioning invokes the Borel-Kolmogorov paradox, a paradox that makes the results of the Bayesian hypothesis test depend on the parametrization used (Consonni & Veronese, 2008; for a summary see Wetzels, Grasman, & Wagenmakers, submitted for publication). Concretely, this means that a test of l ¼ 0 can yield a result that differs from a conceptually equivalent test of l=r ¼ 0.…”

“…These alternatives define priors for nested models so that one does not condition on an event of probability zero. Unfortunately, all methods appear to come with drawbacks of their own (reviewed in Consonni & Veronese, 2008), and presently their does not appear to be a single method that is universally accepted. For most models used in psychology (e.g., regression) the choice of parametrization is clear, but this only slightly alleviates the general concern.…”

Bayesian model comparison based on expected posterior priors for discrete decomposable graphical models

“…This leads us straight into the area of compatible priors, see e.g. Dawid & Lauritzen (2001) and Consonni & Veronese (2008). In this connection, the Expected Posterior Prior (EPP) methodology of Pérez & Berger (2002) represents a useful tool.…”