Inverse Gamma Distribution

The inverse gamma distribution is a continuous probability distribution commonly used as a conjugate prior in Bayesian statistics, particularly for the variance parameter in normal distributions.

Definition

The inverse gamma distribution is a two-parameter continuous probability distribution defined on the positive real line. It is parameterized by a shape parameter α (alpha) and a scale parameter β (beta), where both parameters must be strictly positive.

The probability density function (PDF) is given by:

f(x)=βαΓ(α)x−α−1e−β/x

The cumulative distribution function (CDF) is:

F(x)=Γ(α,β/x)Γ(α)

where Γ(α) is the gamma function and Γ(α,z) is the upper incomplete gamma function.

Applications

  • Bayesian statistics uses the inverse gamma distribution as a conjugate prior for the variance in normal distributions.

  • Reliability theory employs this distribution in lifetime modeling and survival analysis.

  • Machine learning applications use it in hierarchical models for variance parameters.

Properties

Statistical Properties
PropertyValue (α>k where k is the moment order)
Meanβα−1,α>1
Modeβα+1,α>1
Varianceβ2(α−1)2(α−2),α>2
Skewness4α−2α−3,α>3
Excess Kurtosis6(5α−11)(α−3)(α−4),α>4
Entropyα+ln⁡(βΓ(α))−(1+α)ψ(α)

Notable properties include:

Relationships to Other Distributions

  • If X follows a gamma distribution with parameters α and β, then 1/X follows an inverse gamma distribution with the same parameters.

  • The inverse gamma distribution is a special case of the generalized inverse gamma distribution.

  • The scaled inverse chi-square distribution is a special case of the inverse gamma distribution.

The Numerics.NET.Statistics.Distributions.InverseGammaDistribution class

The inverse gamma distribution is implemented by the InverseGammaDistribution class. It has just one constructor with two arguments. The first argument is the shape parameter. The second argument is the scale parameter.

The following constructs an inverse gamma distribution of order 4.2 and scale parameter 1:

C# 
var invGamma = new InverseGammaDistribution(4.2, 1.0);

The InverseGammaDistribution class has two specific properties, ShapeParameter and ScaleParameter, which return the shape and scale parameters of the distribution.

References

For more information on the inverse gamma distribution, refer to the following sources:

  • Johnson, N. L., Kotz, S., & Balakrishnan, N. (1994). "Continuous Univariate Distributions, Volume 1", Chapter 17. Wiley Series in Probability and Statistics.

  • Forbes, C., Evans, M., Hastings, N., & Peacock, B. (2011). "Statistical Distributions", Chapter 22. Wiley Series in Probability and Statistics.

  • Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian Data Analysis. CRC press.

  • Robert, C. P. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer Science & Business Media.

See Also