# Normal-inverse-gamma distribution

Jump to: navigation, search
Parameters $\mu\,$ location (real) $\lambda > 0\,$ (real) $\alpha > 0\,$ (real) $\beta > 0\,$ (real) $x \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)$ $\frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} } \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} e^{ -\frac { 2\beta + \lambda(x - \mu)^2} {2\sigma^2} }$

In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-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

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

has a normal distribution with mean $\mu$ and variance $\sigma^2 / \lambda$, where

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

has an inverse gamma distribution. Then $(x,\sigma^2)$ has a normal-inverse-gamma distribution, denoted as

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

## Characterization

### Probability density function

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

#### Alternative parameterization

It is also possible to let $\gamma = 1 / \lambda$ in which case the pdf becomes

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

## Posterior distribution of the parameters

See the articles on normal-gamma distribution and conjugate prior.

## Interpretation of the parameters

See the articles on normal-gamma distribution and conjugate prior.

## Generating normal-inverse-gamma random variates

Generation of random variates is straightforward:

1. Sample $\sigma^2$ from an inverse gamma distribution with parameters $\alpha$ and $\beta$
2. Sample $x$ from a normal distribution with mean $\mu$ and variance $\sigma^2/\lambda$

## 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.

Content from Wikipedia, the Free Encyclopedia

What Is This Site? The Ultimate Study Guide is a mirror of English Wikipedia. It exists in order to provide Wikipedia content to those who are unable to access the main Wikipedia site due to draconian government, employer, or school restrictions. The site displays all the text content from Wikipedia. Our sponsors generously cover part of the cost of hosting this site, and their ads are shown as part of this agreement. We regret that we are unable to display certain controversial images on some pages the site at the request of the sponsors. If you need to see images which we are unable to show, we encourage you to view Wikipedia directly if possible, and apologize for this inconvenience.

A product of XPR Content Systems. 47 Union St #9K, Grand Falls-Windsor NL A2A 2C9 CANADA