# Inverse Gamma distribution

## Story

If $$Y$$ is Gamma distributed, then $$1/Y$$ is Inverse Gamma distributed.

## Parameters

The number of arrivals, $$\alpha$$, and the rate of arrivals, $$\beta$$.

## Support

The Inverse Gamma distribution is supported on the set of positive real numbers.

## Probability density function

\begin{align} f(y;\alpha, \beta) = \frac{1}{\Gamma(\alpha)}\,\frac{\beta^\alpha}{y^{\alpha+1}} \,\mathrm{e}^{-\beta / y}. \end{align}

## Moments

Mean: $$\displaystyle{\frac{\beta}{\alpha - 1}}$$ for $$\alpha > 1$$; for $$\alpha \le 1$$, the mean is undefined.

Variance: $$\displaystyle{\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}}$$ for $$\alpha > 2$$; for $$\alpha \le 2$$, the variance is undefined.

## Usage

Package

Syntax

NumPy

1 / rg.gamma(alpha, 1/beta)

SciPy

scipy.stats.invgamma(alpha, loc=0, scale=beta)

Stan

inv_gamma(alpha, beta)

## Notes

• The Inverse Gamma distribution is useful as a prior for positive parameters. It imparts a quite heavy tail and keeps probability further from zero than the Gamma distribution.

• NumPy module does not have a function to sample directly from the Inverse Gamma distribution, but it can be achieved by sampling out of a Gamma distribution and then taking the inverser, as shown in the NumPy usage above.