Solved – Gamma parameterization and how to randomly generate $\sigma$’s for use in `rnorm(n, $\mu$, $\sigma$)`

Question

Say I have a normal distribution parameterized with a mean ($\mu$) and precision ($\tau = 1/\sigma^2)$. In JAGS, I would specify a prior for $\tau$ as tau ~ dgamma(0.01, 0.01). Working in R, I want to randomly generate values that would follow the distribution that I've specified in JAGS.

I know that rgamma is the function for drawing values from the gamma distribution. I also know that JAGS parameterizes values differently from R.

In R, the gamma density function is parameterized as $$f(x)= 1/(s^a \Gamma(a)) x^{a-1} e^{-x/s}$$ where $a$ is the shape argument in rgamma, and $s$ is the scale argument in rgamma.

In JAGS, the gamma density function is parameterized as $$\frac{\lambda^rx^{r-1}e^{-\lambda x}}{\Gamma(r)}$$ where $\lambda>0$ and $r>0$.

I think I keep screwing up the algebra with the different parameterizations, and I'm getting confused as to what values I should specify for the arguments of rgamma in R.

I think the correct answer is rgamma(n=1, shape=0.01, rate=1/0.01); is this the correct way to generate a single value from the same ~ dgamma(0.01, 0.01) distribution I specified in JAGS? The greater idea here is to pick a reasonable distribution/ function in R for generating precision values.

Answer 1

The gamma density function in R,

$$f_\mathrm R(x)= 1/(s^a \Gamma(a)) x^{a-1} e^{-x/s},$$

where $a$ is the shape parameter and $s$ is the scale parameter, corresponds to the the dgamma function in JAGS,

$$f_\mathrm{JAGS}(x) = \frac{\lambda^rx^{r-1}e^{-\lambda x}}{\Gamma(r)},$$

where $r$ is the r paramater and $\lambda$ is the lambda parameter,

when we set

$$a = r, s = \frac{1}{\lambda}.$$

If you use these parameters in the R formula, you can see the correspondence:

$$f_\mathrm R(x)= 1\bigg/\left(\left(\frac{1}{\lambda}\right)^r \Gamma(r)\right) x^{r-1} e^{-\lambda x}$$

$$= \frac{\lambda^rx^{r-1}e^{-\lambda x}}{\Gamma(r)}$$

$$= f_\mathrm{JAGS}(x).$$

Hence, the equivalent to a random variable distributed ~ dgamma(0.01, 0.01) in JAGS is rgamma(n = 1, shape = 0.01, scale = 1 / 0.01) in R. The latter one is equivalent to rgamma(n = 1, shape = 0.01, rate = 0.01).