Normal-inverse-gamma distribution

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normal-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-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.

Contents

1 Definition
2 Characterization
- 2.1 Probability density function
  - 2.1.1 Alternative parameterization
- 2.2 Cumulative distribution function
3 Properties
4 Maximum likelihood estimation
5 Posterior distribution of the parameters
6 Interpretation of the parameters
7 Generating normal-gamma random variates
8 Related distributions
9 References

Definition

Suppose

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

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

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

has an inverse gamma distribution. Then $(\mu,\sigma^2)$ has a normal-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 $\gamma = 1 / \nu$ 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

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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-gamma random variates

Generation of random variates is straightforward:

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

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 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.

Probability distributions

Discrete univariate with finite support

Benford Bernoulli Beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf-Mandelbrot

Discrete univariate with infinite support

beta negative binomial Boltzmann Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval, e.g. [0,1]

Arcsine ARGUS Balding-Nichols Bates Beta Beta rectangular Irwin–Hall Kumaraswamy logit-normal Noncentral beta raised cosine triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval, usually [0,∞)

Continuous univariate supported on the whole real line (−∞, ∞)

Cauchy exponential power Fisher's z generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Landau Laplace Linnik logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel variance-gamma Voigt

Continuous univariate with support whose type varies

generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential shifted log-logistic

Mixed continuous-discrete univariate distributions

rectified Gaussian

Multivariate (joint)

Discrete Ewens multinomial multivariate Pólya negative multinomial Continuous Dirichlet Generalized Dirichlet multivariate normal Multivariate stable multivariate Student normal-scaled inverse gamma normal-gamma Matrix-valued inverse-Wishart matrix normal Wishart

Directional

Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular

Degenerate discrete degenerate Dirac delta function Singular Cantor

Families

Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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Normal-inverse-gamma distribution