Is it valid to compare mariginal probability distributions from separate Bayesian analyses to infer which scenario is most likely?
Specifically, in phylogenetic (evolutionary) analysis, if I construct three separate Bayesian analyses with different priors that affect how the phylogenetic tree looks, and each produces a different posterior mariginal probability distribution for the tree, is it valid to compare these tree likelihoods to infer which is the most likely tree topology?
Further, is it valid to use a statistical test to put a P-value on this? i.e. to say the difference between likelihoods is significantly different from zero.
Edit: I think that the results from the normal MCMC runs might not be good for comparisons, instead the mariginal likelihood estimate should be used, e.g. from http://www.beast2.org/path-sampling/:
"Say, the marginal likelihood estimate value for model 1 is X and the value for model 2 is Y, then the Bayes factor comparing model s1 and 2 is is X-Y. If this difference is positive, then the Bayes factor is in favour of model 1, if it is negative, it is in favour of model 2."
If I was to run this many times, then is it possible to calculate a p-value for the comparison? Or, is it better just to say, as worded above, that X or Y model is favoured (and the Bayes factor itself says how much it is favoured). Although this is valid for comparison, and I can now say one is favoured more than the other, I say to use a p-value as then I can say the difference between the mariginal likelihood estimate distributions is significant - so perhaps you can see my thinking, although I am aware this probably makes no sense, it would be great to hear your thoughts!