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Normal-scaled inverse gamma distribution

 
Wikipedia: Normal-scaled inverse gamma distribution
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Normal-scaled inverse gamma
Probability density function
Cumulative distribution function
Parameters \lambda\, location (real)
\nu > 0\, (real)
\alpha > 0\, (real)
\beta > 0\, (real)
Support \mu \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)
Probability density function (pdf) \frac {\sqrt{\nu}} {\sigma\sqrt{2\pi} } \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} e^{ -\frac { 2\beta + \nu(\mu - \lambda)^2} {2\sigma^2} }
Cumulative distribution function (cdf)
Mean
Median
Mode
Variance
Skewness
Excess kurtosis
Entropy
Moment-generating function (mgf)
Characteristic function

In probability theory and statistics, the normal-scaled inverse gamma distribution is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance.

Contents

Definition

Suppose

\mu | \sigma^2, \lambda, \nu \sim \mathrm{N}(\lambda,\sigma^2 / \nu) \,\!

has a normal distribution with mean λ and variance σ2 / ν, where

\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!

has an inverse gamma distribution. Then (μ,σ2) has a normal-scaled inverse gamma distribution, denoted as

(\mu,\sigma^2) \sim \text{N-}\Gamma^{-1}(\lambda,\nu,\alpha,\beta) \! .

Characterization

Probability density function

f(\mu,\sigma^2|\lambda,\nu,\alpha,\beta) = \frac {\sqrt{\nu}} {\sigma\sqrt{2\pi} } \, \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} \exp \left( -\frac { 2\beta + \nu(\mu - \lambda)^2} {2\sigma^2} \right)

Alternative Parameterization

It is also possible to let γ = 1 / ν in which case the pdf becomes

f(\mu,\sigma^2|\lambda,\gamma,\alpha,\beta) = \frac {1} {\sigma\sqrt{2\pi\gamma} } \, \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} \exp \left( -\frac{2\gamma\beta + (\mu - \lambda)^2}{2\gamma \sigma^2} \right)

Cumulative distribution function

Properties

Summation

Scaling

Exponential family

Information entropy

Kullback-Leibler divergence

Maximum likelihood estimation

Generating normal-gamma random variates

Generation of random variates is straightforward:

  1. Sample σ2 from an inverse gamma distribution with parameters α and β
  2. Sample μ from a normal distribution with mean λ and variance σ2 / ν

Related distributions

  • The normal-gamma distribution is the same distribution parameterized by precision rather than variance
  • A generalization of this distribution which allows for a multivariate mean and a positive-definite covariance matrix is the Multivariate normal-inverse Wishart distribution

References

v • d • e
Probability distributions
 
Multivariate (joint)

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Wikipedia. This article is licensed under the Creative Commons Attribution/Share-Alike License. It uses material from the Wikipedia article "Normal-scaled inverse gamma distribution".  Read more