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 x > 0 {\displaystyle x>0}
f ( x ; α , β ) = β α Γ ( α ) ( 1 / x ) α + 1 exp ( − β / x ) {\displaystyle f(x;\alpha ,\beta )={\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}(1/x)^{\alpha +1}\exp \left(-\beta /x\right)}
with shape parameter α {\displaystyle \alpha } and scale parameter β {\displaystyle \beta } . Here Γ ( ⋅ ) {\displaystyle \Gamma (\cdot )} denotes the gamma function. Unlike the gamma distribution, which contains a somewhat similar exponential term, β {\displaystyle \beta } is a scale parameter as the density function satisfies:
f ( x ; α , β ) = f ( x / β ; α , 1 ) β {\displaystyle f(x;\alpha ,\beta )={\frac {f(x/\beta ;\alpha ,1)}{\beta }}}
Cumulative distribution function The cumulative distribution function is the regularized gamma function
F ( x ; α , β ) = Γ ( α , β x ) Γ ( α ) = Q ( α , β x ) {\displaystyle 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 direct computation of Q {\displaystyle Q} , the regularized gamma function.
Moments Provided that α > n {\displaystyle \alpha >n} , the n {\displaystyle n} -th moment of the inverse gamma distribution is given by
E [ X n ] = β n Γ ( α − n ) Γ ( α ) = β n ( α − 1 ) ⋯ ( α − n ) . {\displaystyle \mathrm {E} [X^{n}]=\beta ^{n}{\frac {\Gamma (\alpha -n)}{\Gamma (\alpha )}}={\frac {\beta ^{n}}{(\alpha -1)\cdots (\alpha -n)}}.}
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