# Inverse Gamma Distribution

The inverse gamma distribution (or inverted gamma distribution) is commonly used for Bayesian analysis. It has the same distribution of the reciprocal of the gamma distribution. The shorthand X~inverted gamma(α, β), or IG(α, β), means that a random variable X has an inverse gamma distribution with positive parameters α and β.

## Inverse gamma distribution properties

There are more than two different parameterizations of the inverse gamma. One common one for the probability density function (PDF) is:

The PDF has two positive parameters (α and β):

1. The shape parameter α > 0 controls the height. In general, the greater the alpha, the taller the PDF; higher values for this parameter will also result in thinner tails.
2. The scale parameter β > 0 controls the spread.
3. Γ (α) is the gamma function.

The expected value of the inverse gamma distribution is 

The variance is:

You may see different formulas depending on which parameterization the author chooses for the PDF. For example, some Bayesians parametrize the distribution as a scaled inverse chi-squared distribution.

The inverse chi-squared is a special case of the inverse gamma distribution with α = ν /2 and β = ½, where ν = degrees of freedom  and a special case of type 5 Pearson distribution.

## Uses

The main use of the inverse gamma distribution is in Bayesian probability as a marginal posterior (a way to summarize uncertain quantities) or as a conjugate prior (a prior is a probability distribution that represents your beliefs about a quantity, without taking any evidence into account). The distribution is also used in machine learning, reliability theory (a general theory about systems failure), and survival analysis .

References

Graph of inverse gamma distribution: IkamusumeFan, <https://creativecommons.org/licenses/by-sa/4.0> CC BY-SA 4.0</a>, via Wikimedia Commons

 Inverted Gamma Distribution. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Invertedgamma.pdf

 Gelman, A. et al. (2003). Bayesian Data Analysis, Second Edition. Taylor & Francis.

 Stephanie Glen. “Inverse Gamma Distribution: Definition, Mean, Variance, PDF” From StatisticsHowTo.com: Elementary Statistics for the rest of us! https://www.statisticshowto.com/inverse-gamma-distribution/

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