Normal-scaled inverse gamma distribution

Normal-scaled inverse gamma

parameters:	$\lambda\,$ location (real) $\nu > 0\,$ (real) $\alpha > 0\,$ (real) $\beta > 0\,$ (real)
support:	$\mu \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)$
pdf:	$\frac {\sqrt{\nu}} {\sigma\sqrt{2\pi} } \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} e^{ -\frac { 2\beta + \nu(\mu - \lambda)^2} {2\sigma^2} }$

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.

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

• Dominici, Francesca; Giovanni Parmigiani, Merlise Clyde (2000-09). "Conjugate Analysis of Multivariate Normal Data with Incomplete Observations". The Canadian Journal of Statistics / La Revue Canadienne de Statistique (The Canadian Journal of Statistics / La Revue Canadienne de Statistique, Vol. 28, No. 3) 28 (3): 533–550. doi:10.2307/3315963. JSTOR 3315963.