The problem in hand is that the prior distribution which I have received from experts (loan recovery data) ranges from 0 to 100%. Thus a beta distribution was assumed. Where as the actual data shows that loan recovery can be more than 100% due to fees and interest charges. Thus the likelihood function does not follow a beta distribution simply because the values may be greater than 100%. Trying to use conjugate prior method to combine distributions to arrive at the posterior distribution. First I thought the beta-beta conjugate prior method will suffice. Rescaling the likelihood function to 0% - 100% does not seem correct. What do I do. Please help
The gamma distribution is conjugate and has support from 0 to $\infty$. There is probably a way to convert beta parameters to gamma parameters analytically, but it might be quicker just to fit your beta prior to a gamma distribution. Here's an R example, assuming your beta prior is $\alpha=2, \beta=5$:
require(MASS) y <- rbeta(10000, 2, 5) hist(y, breaks=31, freq=FALSE) fit <- fitdistr(y, "gamma") fit x <- seq(0, 1, 0.01) points(x, dgamma(x, fit$estimate, fit$estimate))
You'll get output like this:
shape rate 2.61543039 9.18351169 (0.03488704) (0.13501986)
You should be able to plug those values into your model with a gamma prior and accommodate values > 1.