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{{Probability distribution |
  name      =normal-inverse-gamma|
  type      =density|
  pdf_image  =|
  cdf_image  =|
  parameters =<math>\mu\,</math> [[location parameter|location]] ([[real number|real]])<br /><math>\lambda > 0\,</math> (real)<br /><math>\alpha > 0\,</math> (real)<br /><math>\beta > 0\,</math> (real)|
  support    =<math>x \in (-\infty, \infty)\,\!, \; \sigma^2 \in (0,\infty)</math>|
  pdf        =<math>\frac {\sqrt{\lambda}} {\sigma\sqrt{2\pi} }  \frac{\beta^\alpha}{\Gamma(\alpha)} \, \left( \frac{1}{\sigma^2} \right)^{\alpha + 1}  e^{ -\frac { 2\beta + \lambda(x - \mu)^2} {2\sigma^2}  } </math>
|
  cdf        =|
  mean      =|
  median    =|
  mode      =|
  variance  =|
  skewness  =|
  kurtosis  =|
  entropy    =|
  mgf        =|
  char      =|
}}
In [[probability theory]] and [[statistics]], the '''normal-inverse-gamma distribution''' (or '''Gaussian-inverse-gamma distribution''') is a four-parameter family of multivariate continuous [[probability distribution]]s. It is the [[conjugate prior]] of a [[normal distribution]] with unknown [[mean]] and [[variance]].
 
==Definition==
Suppose
 
:<math>  x | \sigma^2, \mu, \lambda\sim \mathrm{N}(\mu,\sigma^2 / \lambda) \,\! </math>
has a [[normal distribution]] with [[mean]] <math> \mu</math> and [[variance]] <math> \sigma^2 / \lambda</math>, where
 
:<math>\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!</math>
has an [[inverse gamma distribution]]. Then <math>(x,\sigma^2) </math>
has a normal-inverse-gamma distribution, denoted as
:<math> (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! .
</math>
 
(<math>\text{NIG}</math> is also used instead of <math>\text{N-}\Gamma^{-1}.</math>)
 
In a multivariate form of the normal-inverse-gamma distribution, <math>  \mathbf{x} | \sigma^2, \boldsymbol{\mu}, \mathbf{V}^{-1}\sim \mathrm{N}(\boldsymbol{\mu},\sigma^2 \mathbf{V}) \,\! </math> -- that is, conditional on <math> \sigma^2 </math>, <math>  \mathbf{x} </math> is a <math> k \times 1 </math> random vector that follows the [[multivariate normal distribution]] with mean <math> \boldsymbol{\mu} </math> and [[covariance matrix | covariance]] <math> \sigma^2\mathbf{V}</math> -- while, as in the univariate case, <math>\sigma^2|\alpha, \beta \sim \Gamma^{-1}(\alpha,\beta) \!</math>.
 
==Characterization==
 
===Probability density function===
 
: <math>f(x,\sigma^2|\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) </math>
 
For the multivariate form where <math>  \mathbf{x} </math> is a <math> k \times 1 </math> random vector,
 
: <math>f(\mathbf{x},\sigma^2|\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)^{k/2 + \alpha + 1}  \exp \left( -\frac { 2\beta + (\mathbf{x} - \boldsymbol{\mu})' \mathbf{V}^{-1} (\mathbf{x} - \boldsymbol{\mu})} {2\sigma^2}  \right). </math>
 
where <math>|\mathbf{V}|</math> is the [[determinant]] of the <math> k \times k </math> [[matrix (mathematics) | matrix]] <math>\mathbf{V}</math>. Note how this last equation reduces to the first form if <math>k = 1</math> so that <math>\mathbf{x}, \mathbf{V}, \boldsymbol{\mu}</math> are [[scalar (mathematics)| scalars]].
 
==== Alternative parameterization ====
It is also possible to let <math> \gamma = 1 / \lambda</math> in which case the pdf becomes
 
: <math>f(x,\sigma^2|\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)</math>
 
In the multivariate form, the corresponding change would be to regard the covariance matrix <math>\mathbf{V}</math> instead of its [[invertible matrix | inverse]] <math>\mathbf{V}^{-1}</math> as a parameter.
 
===Cumulative distribution function===
 
==Properties==
 
===Marginal distributions===
 
Given <math> (x,\sigma^2) \sim \text{N-}\Gamma^{-1}(\mu,\lambda,\alpha,\beta) \! .
</math> as above, <math>\sigma^2</math> by itself follows an [[inverse gamma distribution]]:
 
:<math>\sigma^2 \sim \Gamma^{-1}(\alpha,\beta) \!</math>
 
while <math> \sqrt{\frac{\beta}{\alpha\lambda}} (x - \mu) </math> follows a [[Student's t-distribution | t distribution]] with <math> 2 \alpha </math> degrees of freedom.
 
In the multivariate case, the marginal distribution of <math>\mathbf{x}</math> is a [[multivariate Student distribution | multivariate t distribution]]:
 
:<math>\mathbf{x} \sim t_{2\alpha}(\boldsymbol{\mu}, \frac{\beta}{\alpha} \mathbf{V}) \!</math>
 
===Summation===
 
===Scaling===
 
===Exponential family===
 
===Information entropy===
 
===Kullback-Leibler divergence===
 
== Maximum likelihood estimation ==
 
{{Empty section|date=July 2010}}
 
== 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:
# Sample <math>\sigma^2</math> from an inverse gamma distribution with parameters <math>\alpha</math> and <math>\beta</math>
# Sample <math>x</math> from a normal distribution with mean <math>\mu</math> and variance <math>\sigma^2/\lambda</math>
 
== Related distributions ==
* The [[normal-gamma distribution]] is the same distribution parameterized by [[precision (statistics)|precision]] rather than [[variance]]
* A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix <math>\sigma^2 \mathbf{V}</math> (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor <math>\sigma^2</math>) 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
 
{{ProbDistributions|multivariate}}
 
[[Category:Continuous distributions]]
[[Category:Multivariate continuous distributions]]
[[Category:Normal distribution]]
[[Category:Probability distributions]]

Revision as of 17:26, 13 September 2013

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|σ2,μ,λN(μ,σ2/λ)

has a normal distribution with mean μ and variance σ2/λ, where

σ2|α,βΓ1(α,β)

has an inverse gamma distribution. Then (x,σ2) has a normal-inverse-gamma distribution, denoted as

(x,σ2)N-Γ1(μ,λ,α,β).

(NIG is also used instead of N-Γ1.)

In a multivariate form of the normal-inverse-gamma distribution, x|σ2,μ,V1N(μ,σ2V) -- that is, conditional on σ2, x is a k×1 random vector that follows the multivariate normal distribution with mean μ and covariance σ2V -- while, as in the univariate case, σ2|α,βΓ1(α,β).

Characterization

Probability density function

f(x,σ2|μ,λ,α,β)=λσ2πβαΓ(α)(1σ2)α+1exp(2β+λ(xμ)22σ2)

For the multivariate form where x is a k×1 random vector,

f(x,σ2|μ,V1,α,β)=|V|1/2(2π)k/2βαΓ(α)(1σ2)k/2+α+1exp(2β+(xμ)V1(xμ)2σ2).

where |V| is the determinant of the k×k matrix V. Note how this last equation reduces to the first form if k=1 so that x,V,μ are scalars.

Alternative parameterization

It is also possible to let γ=1/λ in which case the pdf becomes

f(x,σ2|μ,γ,α,β)=1σ2πγβαΓ(α)(1σ2)α+1exp(2γβ+(xμ)22γσ2)

In the multivariate form, the corresponding change would be to regard the covariance matrix V instead of its inverse V1 as a parameter.

Cumulative distribution function

Properties

Marginal distributions

Given (x,σ2)N-Γ1(μ,λ,α,β). as above, σ2 by itself follows an inverse gamma distribution:

σ2Γ1(α,β)

while βαλ(xμ) follows a t distribution with 2α degrees of freedom.

In the multivariate case, the marginal distribution of x is a multivariate t distribution:

xt2α(μ,βαV)

Summation

Scaling

Exponential family

Information entropy

Kullback-Leibler divergence

Maximum likelihood estimation

Template:Empty section

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 σ2 from an inverse gamma distribution with parameters α and β
  2. Sample x from a normal distribution with mean μ and variance σ2/λ

Related distributions

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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