Dimitris Fouskakis, Ioannis Ntzoufras and Konstantinos Perrakis, from Athens, have just arXived a paper on power-expected-posterior priors. Just like the power prior and the expected-posterior prior, this approach aims at avoiding improper priors by the use of imaginary data, which distribution is itself the marginal against another prior. (In the papers I wrote on that topic with Juan Antonio Cano and Diego Salmerón, we used MCMC to figure out a fixed point for such priors.)
The current paper (which I only perused) studies properties of two versions of power-expected-posterior priors proposed in an earlier paper by the same authors. For the normal linear model. Using a posterior derived from an unormalised powered likelihood either (DR) integrated in the imaginary data against the prior predictive distribution of the reference model based on the powered likelihood, or (CR) integrated in the imaginary data against the prior predictive distribution of the reference model based on the actual likelihood. The baseline model being the G-prior with g=n². Both versions lead to a marginal likelihood that is similar to BIC and hence consistent. The DR version coincides with the original power-expected-posterior prior in the linear case. The CR version involves a change of covariance matrix. All in all, the CR version tends to favour less complex models, but is less parsimonious as a variable selection tool, which sounds a wee bit contradictory. Overall, I thus feel (possibly incorrectly) that the paper is more an appendix to the earlier paper than a paper in itself as I do not get in the end a clear impression of which method should be preferred.