Inverse-gamma distribution

Inverse-gamma

Probability density function 325px
Cumulative distribution function 325px
Parameters	$\alpha>0$ shape (real) $\beta>0$ 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 $\alpha > 1$
Mode	$\frac{\beta}{\alpha+1}\!$
Variance	$\frac{\beta^2}{(\alpha-1)^2(\alpha-2)}\!$ for $\alpha > 2$
Skewness	$\frac{4\sqrt{\alpha-2}}{\alpha-3}\!$ for $\alpha > 3$
Ex. kurtosis	$\frac{30\,\alpha-66}{(\alpha-3)(\alpha-4)}\!$ for $\alpha > 4$
Entropy	$\alpha\!+\!\ln(\beta\Gamma(\alpha))\!-\!(1\!+\!\alpha)\Psi(\alpha)$
MGF	Does not exist.
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 the distribution arises as the marginal posterior distribution for the unknown variance of a normal distribution if an uninformative prior is used; and as an analytically tractable conjugate prior if an informative prior is required.

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. Other Bayesians prefer to parametrize the inverse gamma distribution differently, as a scaled inverse chi-squared distribution.

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 $\alpha$ and scale parameter $\beta$. Here $\Gamma(\cdot)$ denotes the gamma function.

Unlike the Gamma distribution, which contains a somewhat similar exponential term, $\beta$ is a scale parameter as the distribution function satisfies:

$f(x; \alpha, \beta) = \frac{f(\frac{x}{\beta}; \alpha, 1)}{\beta}$

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.

Characteristic function

$K_{\alpha}(\cdot)$ in the expression of the characteristic function is the modified Bessel function of II kind.

Properties

For $\alpha>0$ and $\beta>0$

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

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

where $\psi(\alpha)$ is the digamma function.

Differential equation

$\left\{x^2 f'(x)+f(x) (-\beta +\alpha x+x)=0,f(1)=\frac{e^{-\beta } \beta ^{\alpha }}{\Gamma (\alpha )}\right\}$

Related distributions

• If $X \sim \mbox{Inv-Gamma}(\alpha, \beta)$ 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 $X \sim \textrm{Inv-Gamma}(\tfrac{1}{2},\tfrac{c}{2})$ then $X \sim \textrm{Levy}(0,c)\,$ (Lévy distribution)
• If $X \sim \mbox{Gamma}(k, \theta)\,$ (Gamma distribution) then $\tfrac{1}{X} \sim \mbox{Inv-Gamma}(k, \theta^{-1})\,$ (see derivation in the next paragraph for details)
• 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.
• For the distribution of a sum of independent inverted Gamma variables see Witkovsky (2001)

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 $\alpha$; $\theta^{-1}$ with $\beta$; 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

References

• V. Witkovsky (2001) Computing the distribution of a linear combination of inverted gamma variables, Kybernetika 37(1), 79-90