I have a stats question that relates to a bigger piece of research, but I've simplified the bit I'm confused about...

Suppose I run an experiment 10 times (M=10) where for each experiment I toss a coin 20 times (n=20), I have for each experiment the number of heads (x). From this its is quite easy to estimate the probability of a success p^hat = sum(x_i/n_i)/M

Suppose in the same experiment I actually swapped the coin half way through, and I have the results for each of the coins' trials. Again the best fit is quite easy to find.

What I would like to do is compare whether assuming the probabilities for both coins are the same. I would like to have separate models (model 1: 20 trials each, model 2: 10 trials for each coin) for each hypothesis and compare hypotheses. But I'm unsure how to do this...usually I would compare the log-likelihood, but the models aren't nested so I don't think this is appropriate. (The values in the combination of the binomial likelihood differ for each model) I could assume that for the same model p1=p2, find the log-likelihood and compare to parameters being unconstrained. This is ok for the parameters but I'm not comparing models, so its not really what I want.

I think I might be able to use Bayes Factors, can anyone comment on this and perhaps provide introductory references? Or is there a simpler solution?