Bayesian Hypothesis Testing: Bayes Factor vs Comparing the Posterior Distribution If we are to choose between two hypotheses $H_0$ vs $H_1$, it seems natural to compare the posterior probabilities $P(H_0|D)$ vs $P(H_1|D)$ and pick the one with the larger probability. In other words, we can look at the posterior probability ratio $\frac{P(H_0|D)}{P(H_1|D)}$.
Yet, we often use the size of the Bayes factor to choose between hypotheses:
$$
\frac{P(D|H_0)}{P(D|H_1)} = \frac{P(H_0|D)}{P(H_1|D)}/\frac{P(H_0)}{P(H_1)}
$$
The two ratios are usually different because usually $P(H_0) \ne P(H_1)$ as
$$
P(H_i) = \int_{\Theta_i} \pi(\theta) d\theta
$$
I think choosing the hypotheses using the two criteria is equivalent to making a decision under $a_0-a_1$ loss with different $a_0, a_1$ values, but, still, I wonder why we would want to choose different $a0, a_1$ values for essentially the same task and why not choosing any other values.
I have seen people use the Bayes factor all the time, yet rarely see people use the posterior probability ratio (although it is mentioned in some textbooks). Why is that?
 A: I think $P(H_i)$ are just your prior belief of the hypotheses, not sure if that is what you try to depict here as I believe the integral you wrote should simply be equal to 1.
For example look at this R package which lets you specify prior odds for hypotheses/models .
I think most often practitioners assume these are equally likely and calculate Bayes factor as simply the posterior ratio .
A: The Bayes factor formalises how the data change the prior ratio, so it measures what the data have to say about the hypotheses in a Bayesian framework, with the (hypothesis) priors "taken out" if you want. People can then use their own priors for making decisions or formalising their overall belief. One can also argue that the "evidence in the data alone" regarding the hypotheses is of interest in itself, even if ultimately posterior probabilities are used to make decisions.
A more problematic explanation is that many people using Bayesian statistics are very shy to specify informative priors, and convincing justifications of priors are hard to find. The Bayes factor lets them get away without doing so (although parameter priors are still needed for composite hypotheses).
