Bayesian analysis problem

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

• Your data are non-negative ratios, so take the logarithm and model on $\mathbb{R}$. A non-parametric method will probably serve you well. You might need to mix in a point mass at 0% (on the original scale, corresponding to $-\infty$ on the log scale). – Cyan Sep 26 '13 at 18:08

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$estimate[1], fit$estimate[2]))


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.

• Dear Eldan: This looks to be a simple solution, but could you please let me know of any supporting documents for the same? i have tried wikipedia and few other docs, but there is no indication of gamma-gamma conjugate prior model. May be I am wrong in understanding the text, please do guide me to some good reading material on the same. Thanks a ton for your help. – Bik Oct 8 '13 at 6:51