Non Normal Distributions > Inverse Gamma Distribution
What is the 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 inverse chi-squared is a special case of the inverse gamma distribution with α = ν /2 and β = ½, where ν = degrees of freedom [1] and a special case of type 5 Pearson distribution.
Important note on parameterizations: Different textbook authors have different ways of showing parameterization for this distribution (this isn’t too unusual, as many distributions can be parameterized in different ways). For example, some authors define the inverse gamma distribution using a shape-scale parameterization, while others may use a shape-rate parameterization. In other words, if you pick up a different textbook or research paper, the formula for the inverse gamma pdf might look different due to a change in how the parameters are defined. Always double-check the definition being used in the context you’re working in.
Properties
Shape-scale parameterization example 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:
- The shape parameter α controls the height. The higher the alpha, the taller the probability density function (PDF); higher values for the shape parameter will also result in thinner tails.
- The scale parameter β controls the spread.
The generalized inverse gamma distribution has two additional parameters:
- The mean, μ. This is always zero in the two-parameter version.
- γ, which controls the concentration near the x-axis. This is always set to 1 in the two-parameter version.
The distribution spreads over the interval μ to ∞. As values for x increase, the function decreases exponentially, giving the inverse gamma distribution very thin tails. One possible parameterization for the inverted gamma distribution has the probability density function (pdf) of: 
The distribution is closely related to the chi square distribution: the pdf of the inverse gamma distribution [ν, 1/2] is the same as the Inverse Chi Square Distribution. The mean (for α > 2) is: E(X) = β / (α – 1). The variance is: β2 / ((α – 1)2*(α – 2)).
Uses of the inverse gamma distribution
The main function of the inverse gamma distribution is in Bayesian probability, where it is used 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). In other words, it’s used to model uncertain quantities. The inverse gamma distribution is also used in machine learning, reliability theory (a general theory about systems failure), and survival analysis.
References
- Gelman, A. et al. (2003). Bayesian Data Analysis, Second Edition. Taylor & Francis.