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 .

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 .