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:

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