Normal-inverse-gamma distribution

normal-inverse-gamma
	Probability density function
Parameters	location (real); (real); (real); (real)
Support
PDF
Mean	; , for ;
Mode	;
Variance	, for ; , for ; , for

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\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 ${\text{N-}}\Gamma ^{-1}.$ )

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

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 }})^{T}\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.^[1]

Proof for $\lambda =1$

For $\lambda =1$ probability density function is

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

Marginal distribution over $x$ is

${\begin{aligned}f(x\mid \mu ,\alpha ,\beta )&=\int _{0}^{\infty }d\sigma ^{2}f(x,\sigma ^{2}\mid \mu ,\alpha ,\beta )\\&={\frac {1}{\sqrt {2\pi }}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\int _{0}^{\infty }d\sigma ^{2}\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1/2+1}\exp \left(-{\frac {2\beta +(x-\mu )^{2}}{2\sigma ^{2}}}\right)\end{aligned}}$

Except for normalization factor, expression under the integral coincides with Inverse-gamma distribution

$\Gamma ^{-1}(x;a,b)={\frac {b^{a}}{\Gamma (a)}}{\frac {e^{-b/x}}{{x}^{a+1}}},$

with $x=\sigma ^{2}$ , $a=\alpha +1/2$ , $b={\frac {2\beta +(x-\mu )^{2}}{2}}$ .

Since $\int _{0}^{\infty }dx\Gamma ^{-1}(x;a,b)=1,\quad \int _{0}^{\infty }dxx^{-(a+1)}e^{-b/x}=\Gamma (a)b^{-a}$ , and

$\int _{0}^{\infty }d\sigma ^{2}\left({\frac {1}{\sigma ^{2}}}\right)^{\alpha +1/2+1}\exp \left(-{\frac {2\beta +(x-\mu )^{2}}{2\sigma ^{2}}}\right)=\Gamma (\alpha +1/2)\left({\frac {2\beta +(x-\mu )^{2}}{2}}\right)^{-(\alpha +1/2)}$

Substituting this expression and factoring dependence on $x$ ,

$f(x\mid \mu ,\alpha ,\beta )\propto _{x}\left(1+{\frac {(x-\mu )^{2}}{2\beta }}\right)^{-(\alpha +1/2)}.$

Shape of generalized Student's t-distribution is

$t(x|\nu ,{\hat {\mu }},{\hat {\sigma }}^{2})\propto _{x}\left(1+{\frac {1}{\nu }}{\frac {(x-{\hat {\mu }})^{2}}{{\hat {\sigma }}^{2}}}\right)^{-(\nu +1)/2}$ .

Marginal distribution $f(x\mid \mu ,\alpha ,\beta )$ follows t-distribution with $2\alpha$ degrees of freedom

$f(x\mid \mu ,\alpha ,\beta )=t(x|\nu =2\alpha ,{\hat {\mu }}=\mu ,{\hat {\sigma }}^{2}=\beta /\alpha )$ .

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} )\!

Summation

Scaling

Suppose

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

Then for $c>0$ ,

(cx,c\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(c\mu ,\lambda /c,\alpha ,c\beta )\!.

Proof: To prove this let $(x,\sigma ^{2})\sim {\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )$ and fix $c>0$ . Defining $Y=(Y_{1},Y_{2})=(cx,c\sigma ^{2})$ , observe that the PDF of the random variable $Y$ evaluated at $(y_{1},y_{2})$ is given by $1/c^{2}$ times the PDF of a ${\text{N-}}\Gamma ^{-1}(\mu ,\lambda ,\alpha ,\beta )$ random variable evaluated at $(y_{1}/c,y_{2}/c)$ . Hence the PDF of $Y$ evaluated at $(y_{1},y_{2})$ is given by : $f_{Y}(y_{1},y_{2})={\frac {1}{c^{2}}}{\frac {\sqrt {\lambda }}{\sqrt {2\pi y_{2}/c}}}\,{\frac {\beta ^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{y_{2}/c}}\right)^{\alpha +1}\exp \left(-{\frac {2\beta +\lambda (y_{1}/c-\mu )^{2}}{2y_{2}/c}}\right)={\frac {\sqrt {\lambda /c}}{\sqrt {2\pi y_{2}}}}\,{\frac {(c\beta )^{\alpha }}{\Gamma (\alpha )}}\,\left({\frac {1}{y_{2}}}\right)^{\alpha +1}\exp \left(-{\frac {2c\beta +(\lambda /c)\,(y_{1}-c\mu )^{2}}{2y_{2}}}\right).\!$

The right hand expression is the PDF for a ${\text{N-}}\Gamma ^{-1}(c\mu ,\lambda /c,\alpha ,c\beta )$ random variable evaluated at $(y_{1},y_{2})$ , which completes the proof.

Exponential family

Normal-inverse-gamma distributions form an exponential family with natural parameters $\textstyle \theta _{1}={\frac {-\lambda }{2}}$ , $\textstyle \theta _{2}=\lambda \mu$ , $\textstyle \theta _{3}=\alpha$ , and $\textstyle \theta _{4}=-\beta +{\frac {-\lambda \mu ^{2}}{2}}$ and sufficient statistics $\textstyle T_{1}={\frac {x^{2}}{\sigma ^{2}}}$ , $\textstyle T_{2}={\frac {x}{\sigma ^{2}}}$ , $\textstyle T_{3}=\log {\big (}{\frac {1}{\sigma ^{2}}}{\big )}$ , and $\textstyle T_{4}={\frac {1}{\sigma ^{2}}}$ .

Information entropy

Kullback–Leibler divergence

Measures difference between two distributions.

Maximum likelihood estimation

Posterior distribution of the parameters

See the articles on normal-gamma distribution and conjugate prior.

Interpretation of the parameters