Normal-inverse-gamma distribution

normal-inverse-gamma
Parameters	$\mu \,$ location (real) $\lambda >0\,$ (real) $\alpha >0\,$ (real) $\beta >0\,$ (real)
Support	$x \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)$
PDF	$\frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} } \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1} e^{ -\frac { 2\beta + \lambda(x - \mu)^2} {2\sigma^2} }$

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) \! .

( $\text{NIG}$ is also used instead of $\text{N-}\Gamma^{-1}.$ )

In a multivariate form of the normal-inverse-gamma distribution, $\mathbf {x} |\sigma ^{2},{\boldsymbol {\mu }},\mathbf {V} ^{-1}\sim \mathrm {N} ({\boldsymbol {\mu }},\sigma ^{2}\mathbf {V} ^{-1})\,\!$ -- that is, conditional on $\sigma ^{2}$ , $\mathbf{x}$ is a $k\times 1$ random vector that follows the multivariate normal distribution with mean ${\boldsymbol {\mu }}$ and covariance $\sigma ^{2}\mathbf {V} ^{-1}$ -- while, as in the univariate case, $\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!$ .

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)

For the multivariate form where $\mathbf{x}$ is a $k\times 1$ random vector,

f(\mathbf{x},\sigma^2|\mu,\mathbf{V}^{-1},\alpha,\beta) = |\mathbf{V}|^{-1/2} {(2\pi)^{-k/2} } \, \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{k/2 + \alpha + 1} \exp \left( -\frac { 2\beta + (\mathbf{x} - \boldsymbol{\mu})' \mathbf{V}^{-1} (\mathbf{x} - \boldsymbol{\mu})} {2\sigma^2} \right).

where $|\mathbf{V}|$ is the determinant of the $k \times k$ matrix $\mathbf {V}$ . Note how this last equation reduces to the first form if $k=1$ so that $\mathbf{x}, \mathbf{V}, \boldsymbol{\mu}$ are scalars.

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)

In the multivariate form, the corresponding change would be to regard the covariance matrix $\mathbf {V}$ instead of its inverse $\mathbf {V} ^{-1}$ as a parameter.

Cumulative distribution function

F(x,\sigma^2|\mu,\lambda,\alpha,\beta) = \frac{e^{-\frac{\beta }{\sigma ^2}} \left(\frac{\beta }{\sigma ^2}\right)^{\alpha } \left(\text{erf}\left(\frac{\sqrt{\lambda } (x-\mu )}{\sqrt{2} \sigma }\right)+1\right)}{2 \sigma ^2 \Gamma (\alpha )}

Differential equation

The probability density function of the normal-inverse-gamma distribution is a solution to the following differential equation:

\left\{{\begin{array}{l}\sigma ^{2}f'(x)+\lambda f(x)(x-\mu )=0,\\f(0)={\frac {{\sqrt {\lambda }}\beta ^{\alpha }\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha -1}e^{\frac {-2\beta -\lambda \mu ^{2}}{2\sigma ^{2}}}}{{\sqrt {2\pi }}\sigma \Gamma (\alpha )}}\end{array}}\right\}

Properties

Marginal distributions

Given $(x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! .$ as above, $\sigma ^{2}$ by itself follows an inverse gamma distribution:

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

while $\sqrt{\frac{\alpha\lambda}{\beta}} (x - \mu)$ follows a t distribution with $2 \alpha$ degrees of freedom.

In the multivariate case, the marginal distribution of $\mathbf {x}$ is a multivariate t distribution:

\mathbf {x} \sim t_{2\alpha }({\boldsymbol {\mu }},{\frac {\beta }{\alpha }}\mathbf {V} ^{-1})\!

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:

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

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 completely unknown positive-definite covariance matrix $\sigma^2 \mathbf{V}$ (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor $\sigma ^{2}$ ) is the normal-inverse-Wishart distribution

References

Denison, David G. T. ; Holmes, Christopher C.; Mallick, Bani K.; Smith, Adrian F. M. (2002) Bayesian Methods for Nonlinear Classification and Regression, Wiley. ISBN 0471490369
Koch, Karl-Rudolf (2007) Introduction to Bayesian Statistics (2nd Edition), Springer. ISBN 354072723X

Probability distributions
List
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 Borel 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	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle
Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda
Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt
Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Marchenko–Pastur q-exponential q-Gaussian q-Weibull shifted log-logistic Tukey lambda
Mixed continuous-discrete univariate	rectified Gaussian
Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate Laplace multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart
Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham
Degenerate and singular	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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