# Gamma distribution - parameterization -- gamma distribution

**URL:** <https://discourse.mc-stan.org/t/gamma-distribution-parameterization-gamma-distribution/2652>\
**Category:** General\
**Created:** [November 30, 2017, 10:54pm UTC](https://discourse.mc-stan.org/t/gamma-distribution-parameterization-gamma-distribution/2652 "2017-11-30T22:54:09Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![CH\_H](https://avatars.discourse-cdn.com/v4/letter/c/bbce88/32.png) [@CH\_H](https://discourse.mc-stan.org/u/CH_H)\
**Post date:** [November 30, 2017, 10:54pm UTC](https://discourse.mc-stan.org/t/gamma-distribution-parameterization-gamma-distribution/2652/1 "2017-11-30T22:54:09Z")

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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]

---

<div class="post-metadata">

**Author:** ![Bob\_Carpenter](https://yyz2.discourse-cdn.com/flex030/user_avatar/discourse.mc-stan.org/bob_carpenter/32/9230_2.png) [@Bob\_Carpenter](https://discourse.mc-stan.org/u/Bob_Carpenter)\
**Post date:** [December 21, 2017, 11:50pm UTC](https://discourse.mc-stan.org/t/gamma-distribution-parameterization-gamma-distribution/2652/2 "2017-12-21T23:50:22Z")

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Sorry for the late reply.

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;

```

P.S. I escaped your code so the post doesn’t look like a ransom note (blame markdown—they like to guess what you mean and apply seemingly random formatting).