Basically all your answers are in Koop’s 2003 textbook “Bayesian Econometrics” on pages 4-5.
- For Bayesian model comparison models do not need to be nested and there is a natural degrees of freedom correction. Hence, as long as you use the same data having different parameters does not matter at all.
- Short answer: that is sufficient. Longer answer: the prior over the parameters does not matter but the prior odds ratio over the models (see Koop (2003), p. 4). If you a priori assign equal probability to all models (0.5 for your two models), you an simply compare the marginal data densities.
- Exactly.
- You want the marginal data density to be a high as possible. The logarithm is a monotonic transformation, so you also want the log marginal densities to be as high as possible.
- No, you don’t. However, it is often easier to compare models using posterior model probabilities, see Koop (2003), p. 4.