Re: conjugate prior of gamma distribution
- From: Lucas <lscharen@xxxxxxxxx>
- Date: Tue, 05 Jun 2007 08:17:03 -0700
On May 30, 8:58 pm, mistersh...@xxxxxxxxx wrote:
what is the conjugate prior of the gamma distribution (with both
parameters unknown)?
Ideally, I am looking for:
- 3 or 4 hyper-parameters that define a probability distribution over
a 2-parameter gamma distribution
- an algorithm to update the hyperparameters given a data point x
- a formula to calculate the value of the pdf at a value x (or if
that's too hard, the pdf over the parameters)
Thank you!!
Neil
Given the sum of the data, S, the product of the data, P, and the
number of data points, n along with the four hyper-parameters p, q, r,
and s, the conjugate prior to a Gamma distribution with unknown alpha
and beta is given as
p( alpha, beta | p, q, r, s ) = (1/K) (p^(alpha-1) * exp(-beta * q))/
(Gamma(alpha)^r * beta^(-alpha*s))
where alpha > 0 and beta > 0 and is zero otherwise.
The posterior distribution's parameters are given by
p' = p * P
q' = q + S
r' = r + n
s' = s + n
This conjugate prior was published by Miller (1980)* and appears in
Section 3.2 of "A Compendium of Conjugate Priors" by Daniel Fink
http://www.people.cornell.edu/pages/df36/Publications.htm
-Lucas
[*] Miller, Robert B. "Bayesian Analysis of the Two-Parameter Gamma
Distribution" Technometrics, 1980, 22(1), 65-69
.
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