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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.
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Inverse gamma random variables
Mode of an Inverse Gamma Distribution
MLEs of an Inverse Gamma Distribution
Mean and Variance of an Inverse Gamma Distribution
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.
Moments
Provided that , the -th moment of the inverse gamma distribution is given by[2]
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.
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.
Hoff, P. (2009). "A first course in bayesian statistical methods". Springer.
Witkovsky, V. (2001). "Computing the Distribution of a Linear Combination of Inverted Gamma Variables". Kybernetika. 37 (1): 79–90. MR1825758. Zbl1263.62022.