Probability density function
Cumulative distribution function
(see digamma function)
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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.
The inverse gamma distribution's probability density function is defined over the support
with shape parameter and scale parameter . Here denotes the gamma function.
Unlike the Gamma distribution, which contains a somewhat similar exponential term, is a scale parameter as the distribution function satisfies:
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. Many math packages allow direct computation of , the regularized gamma function.
Provided that , the -th moment of the inverse gamma distribution is given by
in the expression of the characteristic function is the modified Bessel function of the 2nd kind.
For and ,
The information entropy is
where 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):
where are the pdfs of the Inverse-Gamma distributions and are the pdfs of the Gamma distributions, is Gamma(αp, βp) distributed.
Let , and recall that the pdf of the gamma distribution is
Note that is the rate parameter from the perspective of the gamma distribution.
Define the transformation . Then, the pdf of is
Note that is the scale parameter from the perspective of the inverse gamma distribution. This can be straightforwardly demonstrated by seeing that satisfies the conditions for being a scale parameter.
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