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Template:Probability 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.

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

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

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

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

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

In a multivariate form of the normal-inverse-gamma distribution, $\mathbf{x}|{\sigma }^2,\mathit{\mu },{\mathbf{V}}^{-1}\sim \mathrm{N}(\mathit{\mu },{\sigma }^2\mathbf{V})$ -- that is, conditional on ${\sigma }^2$, $\mathbf{x}$ is a $k\times 1$ random vector that follows the multivariate normal distribution with mean $\mathit{\mu }$ and covariance ${\sigma }^2\mathbf{V}$ -- while, as in the univariate case, ${\sigma }^2|\alpha ,\beta \sim {\mathrm{\Gamma }}^{-1}(\alpha ,\beta )$.

Characterization

Probability density function

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

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

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

where $\left|\mathbf{V}\right|$ 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 $\mathbf{x},\mathbf{V},\mathit{\mu }$ are scalars.

Alternative parameterization

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

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

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.

Properties

Marginal distributions

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

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

while $\sqrt{\frac{\beta }{\alpha \lambda }}(x-\mu )$ 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:

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

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

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