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

Inprobability theory andstatistics, thenormal-inverse-gamma distribution (orGaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuousprobability distributions. It is theconjugate prior of anormal distribution with unknownmean andvariance.

Suppose

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

has anormal distribution withmean${\displaystyle \mu }$ andvariance${\displaystyle {\sigma }^2/\lambda }$, where

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

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

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

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

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

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

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

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

where${\displaystyle |\mathbf{V}|}$ is thedeterminant 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${\displaystyle \mathbf{x},\mathbf{V},\mathit{\mu }}$ arescalars.

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

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

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

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

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

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

while${\displaystyle \sqrt{\frac{\alpha \lambda }{\beta }}(x-\mu )}$ follows at distribution with${\displaystyle 2\alpha }$ degrees of freedom.^[1]

Proof for${\displaystyle \lambda =1}$

For${\displaystyle \lambda =1}$ probability density function is

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

Marginal distribution over${\displaystyle x}$ is

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

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

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

Since${\displaystyle {\int }_0^{\mathrm{\infty }}dx{\mathrm{\Gamma }}^{-1}(x;a,b)=1,{\int }_0^{\mathrm{\infty }}dx{x}^{-(a+1)}{e}^{-b/x}=\mathrm{\Gamma }(a)b^{-a}}$, and

${\displaystyle {\int }_0^{\mathrm{\infty }}d{\sigma }^2{\left(\frac{1}{{\sigma }^2}\right)}^{\alpha +1/2+1}\exp \left(-\frac{2\beta +(x-\mu {)}^2}{2{\sigma }^2}\right)=\mathrm{\Gamma }(\alpha +1/2){\left(\frac{2\beta +(x-\mu {)}^2}{2}\right)}^{-(\alpha +1/2)}}$

Substituting this expression and factoring dependence on${\displaystyle x}$,

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

Shape ofgeneralized Student's t-distribution is

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

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

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

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

${\displaystyle \mathbf{x}\sim t_{2\alpha }(\mathit{\mu },\frac{\beta }{\alpha }\mathbf{V})}$

Suppose

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

Then for${\displaystyle c>0}$,

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

Proof: To prove this let${\displaystyle (x,{\sigma }^2)\sim \text{N-}{\mathrm{\Gamma }}^{-1}(\mu ,\lambda ,\alpha ,\beta )}$ and fix${\displaystyle c>0}$. Defining${\displaystyle Y=(Y_1,Y_2)=(cx,c{\sigma }^2)}$, observe that the PDF of the random variable${\displaystyle Y}$ evaluated at${\displaystyle (y_1,y_2)}$ is given by${\displaystyle 1/c^2}$ times the PDF of a${\displaystyle \text{N-}{\mathrm{\Gamma }}^{-1}(\mu ,\lambda ,\alpha ,\beta )}$ random variable evaluated at${\displaystyle (y_1/c,y_2/c)}$. Hence the PDF of${\displaystyle Y}$ evaluated at${\displaystyle (y_1,y_2)}$ is given by :${\displaystyle f_Y(y_1,y_2)=\frac{1}{c^2}\frac{\sqrt{\lambda }}{\sqrt{2\pi y_2/c}}\frac{{\beta }^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\left(\frac{1}{y_2/c}\right)}^{\alpha +1}\exp \left(-\frac{2\beta +\lambda (y_1/c-\mu {)}^2}{2y_2/c}\right)=\frac{\sqrt{\lambda /c}}{\sqrt{2\pi y_2}}\frac{(c\beta {)}^{\alpha }}{\mathrm{\Gamma }(\alpha )}{\left(\frac{1}{y_2}\right)}^{\alpha +1}\exp \left(-\frac{2c\beta +(\lambda /c)(y_1-c\mu {)}^2}{2y_2}\right).}$

The right hand expression is the PDF for a${\displaystyle \text{N-}{\mathrm{\Gamma }}^{-1}(c\mu ,\lambda /c,\alpha ,c\beta )}$ random variable evaluated at${\displaystyle (y_1,y_2)}$, which completes the proof.

Normal-inverse-gamma distributions form anexponential family withnatural parameters${\displaystyle {\theta }_1=\frac{-\lambda }{2}}$,${\displaystyle {\theta }_2=\lambda \mu }$,${\displaystyle {\theta }_3=\alpha }$, and${\displaystyle {\theta }_4=-\beta +\frac{-\lambda {\mu }^2}{2}}$ and sufficient statistics${\displaystyle T_1=\frac{x^2}{{\sigma }^2}}$,${\displaystyle T_2=\frac{x}{{\sigma }^2}}$,${\displaystyle T_3=\log (\frac{1}{{\sigma }^2})}$, and${\displaystyle T_4=\frac{1}{{\sigma }^2}}$.

Measures difference between two distributions.

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See the articles onnormal-gamma distribution andconjugate prior.

See the articles onnormal-gamma distribution andconjugate prior.

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${\displaystyle {\sigma }^2/\lambda }$

• Thenormal-gamma distribution is the same distribution parameterized byprecision rather thanvariance
• 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 thenormal-inverse-Wishart distribution

• Compound probability distribution

• Denison, David G. T.; et al. (2002).Bayesian Methods for Nonlinear Classification and Regression. Wiley.ISBN 0471490369.
• Koch, Karl-Rudolf (2007).Introduction to Bayesian Statistics (2nd ed.). Springer.ISBN 354072723X.

• Continuous distributions
• Multivariate continuous distributions
• Normal distribution

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