Inverse Gamma Distribution


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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.

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

Graph of the inverse gamma distribution PDF for varying values of α and β.

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 probability density function (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 [1]

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 [2] 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 [3].

References

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

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

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

[3] 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/