Inverse-gamma distribution

Inverse-gamma

Probability density function
Cumulative distribution function
Parameters	α > 0 shape (real) β > 0 scale (real)
Support	$x\in(0;\infty)\!$
PDF	$\frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha - 1} \exp \left(\frac{-\beta}{x}\right)$
CDF	$\frac{\Gamma(\alpha,\beta/x)}{\Gamma(\alpha)} \!$
Mean	$\frac{\beta}{\alpha-1}\!$ for α > 1
Mode	$\frac{\beta}{\alpha+1}\!$
Variance	$\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!$ for α > 2
Skewness	$\frac{4\sqrt{\alpha-2}}{\alpha-3}\!$ for α > 3
Ex. kurtosis	$\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!$ for α > 4
Entropy	$\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)$
MGF	$\frac{2\left(-\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4\beta t}\right)$
CF	$\frac{2\left(-i\beta t\right)^{\!\!\frac{\alpha}{2}}}{\Gamma(\alpha)}K_{\alpha}\left(\sqrt{-4i\beta t}\right)$

In probability theory and statistics, the inverse gamma distribution is a two-parameter family of continuous probability distributions on the positive real line, which is the distribution of the reciprocal of a variable distributed according to the gamma distribution. Perhaps the chief use of the inverse gamma distribution is in Bayesian statistics, where it serves as the conjugate prior of the variance of a normal distribution. However, it is common among Bayesians to consider an alternative parametrization of the normal distribution in terms of the precision, defined as the reciprocal of the variance, which allows the gamma distribution to be used directly as a conjugate prior.

Characterization

Probability density function

The inverse gamma distribution's probability density function is defined over the support x > 0

$f(x; \alpha, \beta) = \frac{\beta^\alpha}{\Gamma(\alpha)} (x)^{-\alpha - 1}\exp\left(-\frac{\beta}{x}\right)$

with shape parameter α and scale parameter β.

Cumulative distribution function

The cumulative distribution function is the regularized gamma function

$F(x; \alpha, \beta) = \frac{\Gamma\left(\alpha,\frac{\beta}{x}\right)}{\Gamma(\alpha)} = Q\left(\alpha, \frac{\beta}{x}\right)\!$

where the numerator is the upper incomplete gamma function and the denominator is the gamma function. Many math packages allow you to compute Q, the regularized gamma function, directly.

Properties

For α > 0 and β > 0

$\mathbb{E}[\ln(X)] = \ln(\beta) - \psi(\alpha).\,$

$\mathbb{E}[X^{-1}] = \frac{\alpha}{\beta}.\,$

where ψ(α) is the digamma function.

Related distributions

• If X∼Inv-Gamma(α,β) then $k X \sim \mbox{Inv-Gamma}(\alpha, k \beta) \,$
• If $X \sim \mbox{Inv-Gamma}(\alpha, \tfrac{1}{2})$ then $X \sim \mbox{Inv-}\chi^2(2 \alpha)\,$ (inverse-chi-squared distribution)
• If $X \sim \mbox{Inv-Gamma}(\tfrac{\alpha}{2}, \tfrac{1}{2})$ then $X \sim \mbox{Scaled Inv-}\chi^2(\alpha,\tfrac{1}{\alpha})\,$ (scaled-inverse-chi-squared distribution)
• If $X \sim \mbox{Gamma}(\alpha, \tfrac{1}{\beta})\,$ (Gamma distribution) then $\tfrac{1}{X} \sim \mbox{Inv-Gamma}(\alpha, \beta)\,$
• If $X \sim \textrm{Levy}(0,c)\,$ (Lévy distribution) then $X \sim \textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})$
• Inverse gamma distribution is a special case of type 5 Pearson distribution
• A multivariate generalization of the inverse-gamma distribution is the inverse-Wishart distribution.

Derivation from Gamma distribution

The pdf of the gamma distribution is

$f(x) = x^{k-1} \frac{e^{-x/\theta}}{\theta^k \, \Gamma(k)}$

and define the transformation $Y = g(X) = \frac{1}{X}$ then the resulting transformation is

$f_Y(y) = f_X \left( g^{-1}(y) \right) \left| \frac{d}{dy} g^{-1}(y) \right|$

$= \frac{1}{\theta^k \Gamma(k)} \left( \frac{1}{y} \right)^{k-1} \exp \left( \frac{-1}{\theta y} \right) \frac{1}{y^2}$

$= \frac{1}{\theta^k \Gamma(k)} \left( \frac{1}{y} \right)^{k+1} \exp \left( \frac{-1}{\theta y} \right)$

$= \frac{1}{\theta^k \Gamma(k)} y^{-k-1} \exp \left( \frac{-1}{\theta y} \right).$

Replacing k with α; θ ^{− 1} with β; and y with x results in the inverse-gamma pdf shown above

$f(x) = \frac{\beta^\alpha}{\Gamma(\alpha)} x^{-\alpha-1} \exp \left( \frac{-\beta}{x} \right).$

See also

• gamma distribution
• inverse-chi-squared distribution
• normal distribution