Suppose I am in the rare situation where I know for certain the prior probability of the null hypothesis. And yes, I mean the marginal probability of the null (not the probability of the null, conditional on the data). Further suppose that I am comparing two groups (i.e., a t-test) and already know the prior probability that my treatment (whatever it is) will work.
Now suppose I want to enter that information into some statistical software that uses Bayesian statistics. To my utter dismay, what do I find but that the software requires me to specify the prior distribution of the parameters, not the null itself.
That is my dilemma.
So, how do I convert a prior probability (in probability metric) into parameter means/standard deviation distributions?
Put differently, given a prior (say p = 0.5 that the treatment "works"), what should the distribution of $\mu_1, \mu_2, \sigma_1, \text{ and } \sigma_2$ look like?