does anyone know any “rule of thumb” that says what difference in marginal data density is great enough to conclude that a model is preferable to the other? Laplace app of marginal data density depends e.g. on prior on the coefficient that constitutes the difference between the two models we want to compare and I am wondering whether small differences in marginal data density are conclusive. Thanks!

Many thanks! I also wondered. Unfortunately, this does not answer your original question about how this depends on prior. Obviously, I can get the same marginal data density using different priors: in one (better) model use priors that are too far from posterior and use wider priors to increase complexity of the model that is penalized. In the second, worse, model, I can work with relatively precise priors, but not too close to posteriors in order to demonstrate that data are informative. As a result, I will get nearly identical marginal data densities.

I would be very grateful if one could help to clarify the issue.