# Normal-inverse-gamma distribution

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.

Parameters Probability density function $\mu \,$ location (real)$\lambda >0\,$ (real)$\alpha >0\,$ (real)$\beta >0\,$ (real) $x\in (-\infty ,\infty )\,\!,\;\sigma ^{2}\in (0,\infty )$ ${\frac {\sqrt {\lambda }}{\sqrt {2\pi \sigma ^{2}}}}{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)$ $\operatorname {E} [x]=\mu$ $\operatorname {E} [\sigma ^{2}]={\frac {\beta }{\alpha -1}}$ , for $\alpha >1$ $x=\mu \;{\textrm {(univariate)}},x={\boldsymbol {\mu }}\;{\textrm {(multivariate)}}$ $\sigma ^{2}={\frac {\beta }{\alpha +1+1/2}}\;{\textrm {(univariate)}},\sigma ^{2}={\frac {\beta }{\alpha +1+k/2}}\;{\textrm {(multivariate)}}$ $\operatorname {Var} [x]={\frac {\beta }{(\alpha -1)\lambda }}$ , for $\alpha >1$ $\operatorname {Var} [\sigma ^{2}]={\frac {\beta ^{2}}{(\alpha -1)^{2}(\alpha -2)}}$ , for $\alpha >2$ $\operatorname {Cov} [x,\sigma ^{2}]=0$ , for $\alpha >1$ ## Definition

Suppose

$x\mid \sigma ^{2},\mu ,\lambda \sim \mathrm {N} (\mu ,\sigma ^{2}/\lambda )\,\!$ has a normal distribution with mean $\mu$ and variance $\sigma ^{2}/\lambda$ , where

$\sigma ^{2}\mid \alpha ,\beta \sim \Gamma ^{-1}(\alpha ,\beta )\!$ 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, $\mathbf {x} \mid \sigma ^{2},{\boldsymbol {\mu }},\mathbf {V} ^{-1}\sim \mathrm {N} ({\boldsymbol {\mu }},\sigma ^{2}\mathbf {V} ^{-1})\,\!$ – 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, $\sigma ^{2}\mid \alpha ,\beta \sim \Gamma ^{-1}(\alpha ,\beta )\!$ .

## Characterization

### Probability density function

$f(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {\sqrt {\lambda }}{\sigma {\sqrt {2\pi }}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (x-\mu )^{2}}{2\sigma ^{2}}}\right)$ For the multivariate form where $\mathbf {x}$ is a $k\times 1$ random vector,

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

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

$F(x,\sigma ^{2}\mid \mu ,\lambda ,\alpha ,\beta )={\frac {e^{-{\frac {\beta }{\sigma ^{2}}}}\left({\frac {\beta }{\sigma ^{2}}}\right)^{\alpha }\left(\operatorname {erf} \left({\frac {{\sqrt {\lambda }}(x-\mu )}{{\sqrt {2}}\sigma }}\right)+1\right)}{2\sigma ^{2}\Gamma (\alpha )}}$ ## 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 $\sigma ^{2}/\lambda$ • 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