Inverse Gamma Distribution#

Univariate, Continuous, Asymmetric, Non-Negative, Heavy-tailed

The Inverse Gamma distribution is a continuous probability distribution defined as the distribution of the reciprocal of a Gamma-distributed random variable. It is characterized by two parameters: the shape parameter \(\alpha\) and the scale parameter \(\beta\).

In mathematics, the Lévy distribution (a special case of the inverse gamma distribution with a shape parameter \(\alpha=0.5\)) describes the hitting time of a Wiener process, which is the probability distribution of the first time this stochastic process reaches a specific level.

In Bayesian statistics, the Inverse Gamma distribution often appears as the marginal posterior distribution for an unknown variance in a normal distribution when using an uninformative prior. It also serves as an analytically tractable conjugate prior when an informative prior is required.

Key properties and parameters#

Support	\(x \in (0, \infty)\)
Mean	\(\dfrac{\beta}{\alpha-1}\) for \(\alpha > 1\)
Variance	\(\dfrac{\beta^2}{(\alpha-1)^2(\alpha - 2)}\) for \(\alpha > 2\)

Parameters:

\(\alpha\) : (float) Shape parameter, \(\alpha > 0\).
\(\beta\) : (float) Scale parameter, \(\beta > 0\).

Alternative parametrization

The Inverse Gamma distribution has 2 alternative parametrizations: in terms of the shape parameter \(\alpha\) and the scale parameter \(\beta\), or in terms of \(\mu\) (mean) and \(\sigma\) (standard deviation).

The link between the parameters is given by:

\[\begin{split} \alpha = \frac{\mu^2}{\sigma^2} + 2 \\ \beta = \frac{\mu^3}{\sigma^2} + \mu \end{split}\]

Probability Density Function (PDF)#

\[ f(x; \alpha, \beta) = \dfrac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha-1}\exp(-\dfrac{\beta}{x}) \]

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

Parameters \(\alpha\) and \(\beta\)

Parameters \(\mu\) and \(\sigma\)

Cumulative Distribution Function (CDF)#

\[ F(x; \alpha, \beta) = \dfrac{\Gamma(\alpha, \dfrac{\beta}{x})}{\Gamma(\alpha)} \]

where \(\Gamma(\alpha, x)\) is the upper incomplete gamma function.