Inverse Gamma distribution

Story

If \(Y\) is Gamma distributed, then \(1/Y\) is Inverse Gamma distributed.

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

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

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

where \(\Gamma(\alpha)\) is the gamma function.

\[\begin{align} F(y;\alpha, \beta) = Q(\alpha, \beta/x), \end{align}\]

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.

Package	Syntax
NumPy	`1 / rng.gamma(alpha, 1/beta)`
SciPy	`scipy.stats.invgamma(alpha, loc=0, scale=beta)`
Distributions.jl	`InverseGamma(alpha, beta)`
Stan	`inv_gamma(alpha, beta)`

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.