Bayes factors with improper priors I have a question regarding model comparison using Bayes factors. In many cases, statisticians are interested on using a Bayesian approach with improper priors (for example some Jeffreys priors and reference priors). 
My question is, in those cases where the posterior distribution of the model parameters is well-defined, is it valid to compare models using Bayes factors under the use of improper priors?
As a simple example consider comparing a Normal model vs. a Logistic model with Jeffreys priors.
 A: No. While improper priors can be okay for parameter estimation under certain circumstances (due to the Bernstein–von Mises theorem), they are a big no-no for model comparison, due to what is known as the marginalization paradox. 
The problem, as the name would suggest, is that the marginal distribution of an improper distribution is not well-defined. Given a likelihood $p_1(x \mid \theta)$ and a prior $p_1(\theta)$: the Bayes factor requires computing the marginal likelihood:
$$p_1(x) = \int_\Theta p_1(x \mid \theta) p_1(\theta) d \theta .$$
If you think of an improper prior as being only known up to proportionality (e.g. $p_1(\theta) \propto 1$), then the problem is that $p_1(x)$ will be multiplied by an unknown constant. In a Bayes factor, you'll be computing the ratio of something with an unknown constant.
Some authors, notably E.T. Jaynes, try to get around this by defining improper priors as the limit of a sequence of proper priors: then the problem is that there may be two different limiting sequences that then give different answers.
