Quote:
Originally Posted by whosnext
This has been discussed before. You can search for "beta prior" here on 2+2.
Formulas exist for a few "simple" cases. If you assume the prior population of a percentage is Beta with mean M and variance V, and you then observe T new observations of which K are "successes", there is a formula that gives you the mean of the posterior distribution.
IIRC a fair amount of algebra is required, but it is doable. I suppose the formula may be given on the internet somewhere as well. If I have some time this afternoon I'll try to re-derive this myself and post what I come up with.
I've done the easy algebra. Others can carry it as far as they'd like. The Beta distribution relies upon two-parameters called A and B. They are related to the mean M and variance V as follows:
M = A/(A+B)
V = (AB)/[((A+B)^2)*(A+B+1)]
Let R=(1-M)/M. If I did the algebra correctly, A and B in terms of R and V:
A = [R-V*((1+R)^2)]/[V*((1+R)^3)]
B= A*R
Then suppose you observe T new observations of which K are successes. Then the posterior distribution is Beta with parameters A' = A+K and B' = B+T-K.
Of course, the mean of the posterior is given by A'/(A'+B') which is easily found. A complete "standalone" formula for the posterior mean can be derived via more algebra if so desired.