# Updating Gamma conjugate prior

I've got some data that looks like it is Gamma distributed. I've constructed the prior distribution from mean=232 and standard deviation = 150, which yield the Gamma distribution parameters:

a_prior (shape) =2.392 and b_prior (scale) =96.98

Now, I'm looking at the Wikipedia page for conjugate priors, and the update rules for the Gamma hyperparameters are:

$$\alpha_0 + n\alpha,\beta_0+\sum_{i=1}^n x_i$$

The mean of my sample data is $278$ and the standard deviation of the sample data is $162$. These yield $a$ and $b$ to be $2.944$ and $94.4$, respectively. The sample data consists of $15$ data points.

I don't understand the update rules from Wikipedia. As I can tell, the posterior a' = prior_a + n * a, which in this case is 2.39 + 15 * 2.944, which would give as the posterior shape parameter 46.49, which seems quite large.

Not to mention the posterior b' parameter, which looks like it's the prior b + sum(data points). My data points add up to 4451, so the posterior b' is:

posterior b' = 96.98 + 4451 = 4548, which, again seems quite large for the scale parameter.

Can someone tell me how the updating for a conjugate Gamma should go? It seems to me that the larger your sample size n is, the more the shape parameter will blow up. Similarly, the more data points you have in the sum, the scale parameter will grow extremely large.

Clearly I am misunderstanding something or then the data is simply not Gamma distributed.

I looked at the Gamma conjugate for Poisson, which contains calculating the equivalent sample size, but I don't see how it relates to updating Gamma prior to Gamma posterior.

Any help on understanding my misunderstanding is appreciated.

The Gamma prior is not conjugate for the Gamma sampling model. See Conjugate prior for a Gamma distribution

The following R code shows how to sample from the posterior of a Gamma model with uniform priors on $[0,1000]$ for the scale and shape parameters, you can play with other priors, using an adaptive MCMC sampler.

set.seed(12345)
# Simulated data
data = rgamma(100,shape = 2.5,scale=97)

# Histogram and likelihood fit
hist(data)
library(MASS)
fitdistr(data,"gamma")

# -log posterior
mlogpost = function(par){
loglik = sum(dgamma(data,shape=par[1],scale = par[2],log=T))
logprior = -2*log(1000)
return(-loglik - logprior)
}

# Library for the MCMC sampler
library(Rtwalk)
# Initial point
init = c(2.5,97)

# Support of the posterior
Support <- function(x) {
((0 < x[1])&(x[2]>0))
}
# Random initial points (see documentation of Rtwalk)
X0 <- function(x) { init + runif(2,-0.1,0.1) }

# Sampler
info <- Runtwalk( dim=2,  Tr=105000,  Obj=mlogpost, Supp=Support, x0=X0(), xp0=X0())

#burn in and thinning the posterior samples
ind = seq(5000,105000,by=10)
sc = info$output[,1][ind] sh = info$output[,2][ind]

# histograms of the posterior samples
hist(sc)
hist(sh)

# summary of the posterior samples
summary(sc)
summary(sh)


The confusion probably comes from the fact that the conjugate prior is for the: $$\Gamma(shape,rate),$$ and not for the otherwise traditionally used: $$\Gamma(shape,scale).$$ The link between the two is that $rate = 1 / scale$.

So a large value of the rate gives a small value of the scale. A large value both in shape and rate means that your Gamma is becoming bell shaped and shrinking, which is what should happen when you have a lot of data.

In your case, the mean of your posterior is: $$E[x|Data] = \frac{\alpha'}{\beta'} = \frac{46.49}{4548} = 0.0102 .$$