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Support x ∈ ( 0 , ∞ ) {displaystyle xin (0,infty )!} PDF β α Γ ( α ) x − α − 1 exp ( − β x ) {displaystyle {rac {eta ^{alpha }}{Gamma (alpha )}}x^{-alpha -1}exp left(-{rac {eta }{x}}ight)} CDF Γ ( α , β / x ) Γ ( α ) {displaystyle {rac {Gamma (alpha ,eta /x)}{Gamma (alpha )}}!} Mean β α − 1 {displaystyle {rac {eta }{alpha -1}}!} for α > 1 {displaystyle alpha >1} Mode β α + 1 {displaystyle {rac {eta }{alpha +1}}!} |
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.
Contents
- Probability density function
- Cumulative distribution function
- Characteristic function
- Properties
- Related distributions
- Derivation from Gamma distribution
- References
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.
Probability density function
The inverse gamma distribution's probability density function is defined over the support
with shape parameter
Unlike the Gamma distribution, which contains a somewhat similar exponential term,
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. Many math packages allow you to compute Q, the regularized gamma function, directly.
Characteristic function
Properties
For
and
where
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
Differential equation:
Related distributions
Derivation from Gamma distribution
The pdf of the gamma distribution with shape parameter α and rate parameter β is
and define the transformation