# Gamma distribution - parameterization -- gamma distribution

#1

Hello, I’m new to stan. I’m wondering how gamma distribution was parameterized. According to the manual, it is (alpha, beta) .
However, my simulation results seem to suggest otherwise. Please focus on the stddev estimates. Please bear with me as beginners, and it is very likely I
Stan Model:

``````data {
int n;                   // sample size
real xbar;               // sample mean
real <lower=0> ss;       // sample sum-of-square
int mID;                 // gamma parameterization: 1 - assume alpha, beta
//                         2 - assumed k, theta
//
}

parameters {
real mu;                 // ped mean
real <lower=0> stddev;   // std dev: ped
}

model {
xbar  ~ normal(mu, stddev/sqrt(n));
if (mID == 1)
ss ~ gamma((n-1.)/2. , (2*stddev^2));
else
ss ~ gamma((n-1.)/2. , 1/(2*stddev^2));
}
``````
``````####################  simulation results  ######################
## Assumed parameter values
N <- 100
MU <- 0
SIGMA <- 0.50

## sample
set.seed(1)
xbar <- rnorm(1, MU, SIGMA/sqrt(N))
ss <- rchisq(1, df=N-1)*(SIGMA^2)

## stan fits (assumed alpha-beta parameterization)
pCall <- list(n=N, ss=ss, xbar=xbar, mID=1)smp <- sampling(mdl, data=pCall)
smp
## Inference for Stan model: ex.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
##          mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
## mu      -0.03    0.00 0.10  -0.23  -0.10  -0.03   0.04   0.16  4000    1
## stddev   0.99    0.00 0.07   0.86   0.94   0.99   1.04   1.13  3544    1
## lp__   -14.68    0.02 0.98 -17.21 -15.09 -14.38 -13.97 -13.70  2032    1

pCall2 <- list(n=N, ss=ss, xbar=xbar, mID=2)
smp2 <- sampling(mdl, data=pCall2)

## stan fits (assumed k-theta parameterization)

## Inference for Stan model: ex.
## 4 chains, each with iter=2000; warmup=1000; thin=1;
## post-warmup draws per chain=1000, total post-warmup draws=4000.
##
##          mean se_mean   sd   2.5%    25%    50%    75%  97.5% n_eff Rhat
## mu      -0.03    0.00 0.05  -0.13  -0.06  -0.03   0.00   0.07  3077    1
## stddev   0.51    0.00 0.04   0.45   0.48   0.51   0.53   0.59  3102    1
## lp__   -14.67    0.03 1.01 -17.45 -15.04 -14.37 -13.96 -13.70  1526    1
``````

[edit: escaped model code]

#2

The manual defines the formulas for the parameterizations we use. In this case, `alpha` is the shape parameter and `beta` the inverse scale. The kernel of the log density for constant parameters is
``````log Gamma(y | a, b) = (a - 1) * y - b * y;