Bayesian analysis problem

Question

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.

Answer 1

Denote $X$ the principal due, $C$ the sum of fees and interest charges, and $R$ the amount recovered.

By construction , $0\le \frac {R}{X+C} \le 1$. So since your actual data contains values higher than unity, it must be the case that each observation is calculated by the ratio $0\le\frac {R}{X} $.

So the question arises, whether your current sample can be considered as a collection of realizations of the same random variables as those reflected in your prior distribution.

The mathematical expression $\frac {R}{X}$ could be verbally described as "Amount recovered as percentage of principal".

The mathematical expression $\frac {R}{X+C}$ could be verbally described as "Amount recovered as percentage of (principal+fees and interest)".

But your prior data could also have measured something like $\frac {R-C}{X}$, which could be verbally described as "Percentage of principal recovered" (this could even obtain negative values, but negatives could have been set to zero).

You need first to clear this up, before proceeding. These are not the same variables, and so your problem is more serious than finding some distribution.