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.

normal-inverse-gamma

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

Definition

Suppose

${\displaystyle x\mid \sigma ^{2},\mu ,\lambda \sim \mathrm {N} (\mu ,\sigma ^{2}/\lambda )\,\!}$

has a normal distribution with mean ${\displaystyle \mu }$ and variance ${\displaystyle \sigma ^{2}/\lambda }$, where

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

has an inverse gamma distribution. Then ${\displaystyle (x,\sigma ^{2})}$ has a normal-inverse-gamma distribution, denoted as

${\displaystyle (x,\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )\!.}$

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

The normal-inverse-Wishart distribution is a generalization of the normal-inverse-gamma distribution that is defined over multivariate random variables.

Characterization

Probability density function

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

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

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

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

Alternative parameterization

It is also possible to let ${\displaystyle \gamma =1/\lambda }$ in which case the pdf becomes

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

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

Cumulative distribution function

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

Properties

Marginal distributions

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

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

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

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

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

Summation

Scaling

Exponential family

Information entropy

Kullback–Leibler divergence

Maximum likelihood estimation

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

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

See also

• Compound probability 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