Inverse-gamma distribution on arvau

Wikipedia / Inverse-gamma distribution

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. It is common among some 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

with shape parameter and scale parameter. denotes the gamma function.

Unlike the gamma distribution is a scale parameter as the density function satisfies:

Cumulative distribution function

The cumulative distribution function is the regularized gamma function

where the numerator is the upper incomplete gamma function and the denominator is the gamma function, the regularized gamma function.

Moments

-th moment of the inverse gamma distribution is given by

Characteristic function

The inverse gamma distribution has characteristic function is the modified Bessel function of the 2nd kind.

Properties

and

The information entropy is

is the digamma function.

The Kullback-Leibler divergence of Inverse-Gamma(αp, βp) from Inverse-Gamma(αq, βq) is the same as the KL-divergence of Gamma(αp, βp) from Gamma(αq, βq):

is Gamma(αp, βp) distributed.

Related distributions

(inverse-chi-squared distribution)
(scaled-inverse-chi-squared distribution)
(Lévy distribution)
(Exponential distribution)
(Gamma distribution with rate (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

, and recall that the pdf of the gamma distribution is

is the rate parameter from the perspective of the gamma distribution.

satisfies the conditions for being a scale parameter.

Occurrence

Hitting time distribution of a Wiener process follows a Lévy distribution.