gamma distribution
There are two forms for the Gamma distribution, each with different definitions for the shape and scale parameters. Rather than asking what the form is used for the gsl_ran_gamma implementation, it's probably easier to ask for the associated definitions for the mean and standard deviation in terms of the shape and scale parameters.
Any pointers to definitions would be appreciated.
I eventually worked out the answer myself (with the help of a mathematician friend).
In JAGS/BUGS we can define the prior distribution on the precision of a normal distribution using a gamma distribution, which also happens to be a conjugate prior for the normal distribution parameterized by precision. We want to be able to specify a gamma prior on the normal distribution using our guess of the mean SD of the normal distribution and the SD of the SD of the normal distribution. In order to do this we need to find the prior distribution that corresponds to the gamma distribution but is the conjugate prior for a normal distribution parameterized by SD.
I found three mentions of this distribution where it is either called the inverted half gamma (Fink, 1997) or the inverted gamma-1 (Adjemian, 2010; LaValle, 1970).
The inverted gamma-1 distribution has two parameters $\nu$ and $s$ which corresponds to $2 \cdot shape$ and $2 \cdot rate$ of the gamma distribution respectively. The mean and SD of the inverted gamma-1 is:
$$ \mu = \sqrt{\frac{s}{2}}\frac{\Gamma(\frac{\nu-1}{2})}{\Gamma(\frac{\nu}{2})} \space \text{ and } \space \sigma^2 = \frac{s}{\nu - 2} - \mu^2$$
It doesn't seem to exist a closed solution that allow us to get $\nu$ and $s$ if we specify $\mu$ and $\sigma$. Adjemian (2010) recommends a numerical approach and fortunately a matlab script that does this is available from the open source platform Dynare. The following is an R translation of that script:
# Copyright (C) 2003-2008 Dynare Team, modified 2012 by Rasmus Bååth
#
# This file is modified R version of an original Matlab file that is part of Dynare.
#
# Dynare is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# Dynare is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
inverse_gamma_specification <- function(mu, sigma) {
sigma2 = sigma^2
mu2 = mu^2
if(sigma^2 < Inf) {
nu = sqrt(2*(2+mu^2/sigma^2))
nu2 = 2*nu
nu1 = 2
err = 2*mu^2*gamma(nu/2)^2-(sigma^2+mu^2)*(nu-2)*gamma((nu-1)/2)^2
while(abs(nu2-nu1) > 1e-12) {
if(err > 0) {
nu1 = nu
if(nu < nu2) {
nu = nu2
} else {
nu = 2*nu
nu2 = nu
}
} else {
nu2 = nu
}
nu = (nu1+nu2)/2
err = 2*mu^2*gamma(nu/2)^2-(sigma^2+mu^2)*(nu-2)*gamma((nu-1)/2)^2
}
s = (sigma^2+mu^2)*(nu-2)
} else {
nu = 2
s = 2*mu^2/pi
}
c(nu=nu, s=s)
}
The R/JAGS script below shows how we can now specify our gamma prior on the precision of a normal distribution.
library(rjags)
model_string <- "model{
y ~ dnorm(0, tau)
sigma <- 1/sqrt(tau)
tau ~ dgamma(shape, rate)
}"
# Here we specify the mean and sd of sigma and get the corresponding
# parameters for the gamma distribution.
mu_sigma <- 100
sd_sigma <- 50
params <- inverse_gamma_specification(mu_sigma, sd_sigma)
shape <- params["nu"] / 2
rate <- params["s"] / 2
data.list <- list(y=NA, shape = shape, rate = rate)
model <- jags.model(textConnection(model_string),
data=data.list, n.chains=4, n.adapt=1000)
update(model, 10000)
samples <- as.matrix(coda.samples(
model, variable.names=c("y", "tau", "sigma"), n.iter=10000))
And now we can check if the sample posteriors (which should mimic the priors as we gave no data to the model, that is, y = NA) are as we specified.
mean(samples[, "sigma"])
## 99.87198
sd(samples[, "sigma"])
## 49.37357
par(mfcol=c(3,1), mar=c(2,2,2,2))
plot(density(samples[, "tau"]), main="tau")
plot(density(samples[, "sigma"]), main="sigma")
plot(density(samples[, "y"]), main="y")
This seems to be correct. Any objections or comments to this method of specifying a prior is much appreciated!
To calculate the shape and rate of the gamma prior as we have done above is not the same as directly using a gamma prior on the SD on a normal distribution. This is illustrated by the R script below.
# Generating random precision values and converting to
# SD using the shape and rate values calculated above
rand_precision <- rgamma(999999, shape=shape, rate=rate)
rand_sd <- 1/sqrt(rand_prec)
# Specifying the mean and sd of the gamma distribution directly using the
# mu and sigma specified before and generating random SD values.
shape2 <- mu^2/sigma^2
rate2 <- mu/sigma^2
rand_sd2 <- rgamma(999999, shape2, rate2)
The two distributions now has the same mean and SD.
mean(rand_sd)
## 99.96195
mean(rand_sd2)
## 99.95316
sd(rand_sd)
## 50.21289
sd(rand_sd2)
## 50.01591
But they are not the same distribution.
plot(density(rand_sd[rand_sd < 400]), col="blue", lwd=4, xlim=c(0, 400))
lines(density(rand_sd2[rand_sd2 < 400]), col="red", lwd=4, xlim=c(0, 400))
From what I've read it seems to be more usual to put a gamma prior on the precision than a gamma prior on the SD. But I don't know what the argument would be for preferring the former over the latter.
Fink, D. (1997). A compendium of conjugate priors. http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.157.5540&rep=rep1&type=pdf
Adjemian, S. (2010). Prior Distributions in Dynare. http://www.dynare.org/stepan/dynare/text/DynareDistributions.pdf
LaValle, I.H. (1970). An introduction to probability, decision, and inference. Holt, Rinehart and Winston New York.
The three parameter gamma distribution is needed only when you need to shift the distribution itself.
In the two-parameter gamma distribution, you could read the shape parameter as a proxy of the most probable value of the distribution, and the scale parameter of how "long" is its tail.
The mean of the distribution is given exactly by the product of the shape and the scale parameters.
If you need to shift the distribution, that's where you use the three-parameters gamma distribution. In this instance, the mean of the distribution is simply $\text{mean}=\text{location}+\text{shape}\cdot \text{scale}$.
Best Answer
If the shape parameter is $k>0$ and the scale is $\theta>0$, one parameterization has density function
$$p(x) = x^{k-1} \frac{ e^{-x/\theta} }{\theta^{k} \Gamma(k)}$$
where the argument, $x$, is non-negative. A random variable with this density has mean $k \theta$ and variance $k \theta^{2}$ (this parameterization is the one used on the wikipedia page about the gamma distribution).
An alternative parameterization uses $\vartheta = 1/\theta$ as the rate parameter (inverse scale parameter) and has density
$$p(x) = x^{k-1} \frac{ \vartheta^{k} e^{-x \vartheta} }{\Gamma(k)}$$
Under this choice, the mean is $k/\vartheta$ and the variance is $k/\vartheta^{2}$.