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, ${\displaystyle \mathbf {x} |\sigma ^{2},{\boldsymbol {\mu }},\mathbf {V} ^{-1}\sim \mathrm {N} ({\boldsymbol {\mu }},\sigma ^{2}\mathbf {V} ^{-1})\,\!}$ -- 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 ${\displaystyle \sigma ^{2}\mathbf {V} ^{-1}}$ -- 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:

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

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:

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

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:

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 $\sigma^2/\lambda$

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