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?

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 .
