I am pretty new to ABC stuff so I may be saying dumb things.

My question is: I ran an ABC with two models $M_1$ and $M_2$ and now I have an approximation of the posterior distribution for both model.

If I do a posterior check by re-runing enough simulation for which I sample the parameters from the posteriors, is it possible to get back the likelihood of both model in order to calculate their Bayes factor?

I was thinking (probably wrongly) that taking the ratio between the number of simulations falling under a small threshold $\epsilon$ may do something related to the Bayes factor, something that may be written like: $$ BF_{1,2}=\frac{P(M1 | d(M1,D) < \epsilon)}{P(M2 | d(M2,D) < \epsilon)}$$ (where $d()$ is the distance function used for the orginal ABC)