Budan's theorem: Difference between revisions

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{{Probability distribution |
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  name      =normal-inverse-Wishart|
  type      =density|
  pdf_image  =|
  cdf_image  =|
  notation =<math>(\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathrm{NIW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu)</math>|
  parameters =<math>\boldsymbol\mu_0\in\mathbb{R}^D\,</math> [[location parameter|location]] (vector of [[real number|real]])<br /><math>\lambda > 0\,</math> (real)<br /><math>\boldsymbol\Psi \in\mathbb{R}^{D\times D}</math> inverse scale matrix ([[positive-definite matrix|pos. def.]])<br /><math>\nu > D-1\,</math> (real)|
  support    =<math>\boldsymbol\mu\in\mathbb{R}^D ; \boldsymbol\Sigma \in\mathbb{R}^{D\times D}</math> [[covariance matrix]] ([[positive-definite matrix|pos. def.]])|
  pdf        =<math>f(\boldsymbol\mu,\boldsymbol\Sigma|\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu) = \mathcal{N}(\boldsymbol\mu|\boldsymbol\mu_0,\tfrac{1}{\lambda}\boldsymbol\Sigma)\ \mathcal{W}^{-1}(\boldsymbol\Sigma|\boldsymbol\Psi,\nu)</math>|
  cdf        =|
  mean      =|
  median    =|
  mode      =|
  variance  =|
  skewness  =|
  kurtosis  =|
  entropy    =|
  mgf        =|
  char      =|
}}
In [[probability theory]] and [[statistics]], the '''normal-inverse-Wishart distribution''' (or '''Gaussian-inverse-Wishart distribution''') is a multivariate four-parameter family of continuous [[probability distribution]]s. It is the [[conjugate prior]] of a [[multivariate normal distribution]] with unknown [[mean]] and [[covariance matrix]] (the inverse of the [[precision matrix]]).<ref name="murphy">Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf]</ref>
 
==Definition==
Suppose
 
:<math>  \boldsymbol\mu|\boldsymbol\mu_0,\lambda,\boldsymbol\Sigma \sim \mathcal{N}\left(\boldsymbol\mu\Big|\boldsymbol\mu_0,\frac{1}{\lambda}\boldsymbol\Sigma\right)</math>
has a [[multivariate normal distribution]] with [[mean]] <math>\boldsymbol\mu_0</math> and [[covariance matrix]] <math>\tfrac{1}{\lambda}\boldsymbol\Sigma</math>, where
 
:<math>\boldsymbol\Sigma|\boldsymbol\Psi,\nu \sim \mathcal{W}^{-1}(\boldsymbol\Sigma|\boldsymbol\Psi,\nu)</math>
has an [[inverse Wishart distribution]]. Then <math>(\boldsymbol\mu,\boldsymbol\Sigma) </math>
has a normal-inverse-Wishart distribution, denoted as
:<math> (\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathrm{NIW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu) .
</math>
 
==Characterization==
 
===Probability density function===
 
: <math>f(\boldsymbol\mu,\boldsymbol\Sigma|\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu) = \mathcal{N}\left(\boldsymbol\mu\Big|\boldsymbol\mu_0,\frac{1}{\lambda}\boldsymbol\Sigma\right) \mathcal{W}^{-1}(\boldsymbol\Sigma|\boldsymbol\Psi,\nu)</math>
 
==Properties==
 
===Scaling===
 
===Marginal distributions===
By construction, the [[marginal distribution]] over <math>\boldsymbol\Sigma</math> is an [[inverse Wishart distribution]], and the [[conditional distribution]] over <math>\boldsymbol\mu</math> given <math>\boldsymbol\Sigma</math> is a [[multivariate normal distribution]]. The [[marginal distribution]] over <math>\boldsymbol\mu</math> is a [[multivariate t-distribution]].
 
== Posterior distribution of the parameters ==
 
{{Empty section|date=March 2013}}
 
== Generating normal-inverse-Wishart random variates ==
Generation of random variates is straightforward:
# Sample <math>\boldsymbol\Sigma</math> from an [[inverse Wishart distribution]] with parameters <math>\boldsymbol\Psi</math> and <math>\nu</math>
# Sample <math>\boldsymbol\mu</math> from a [[multivariate normal distribution]] with mean <math>\boldsymbol\mu_0</math> and variance <math>\boldsymbol \tfrac{1}{\lambda} \boldsymbol\Sigma</math>
 
== Related distributions ==
* The [[normal-Wishart distribution]] is essentially the same distribution parameterized by precision rather than variance. If <math> (\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathrm{NIW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu)</math> then <math>(\boldsymbol\mu,\boldsymbol\Sigma^{-1}) \sim \mathrm{NW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi^{-1},\nu)</math> .
* The [[normal-inverse-gamma distribution]] is the one-dimensional equivalent.
* The [[multivariate normal distribution]] and [[inverse Wishart distribution]] are the component distributions out of which this distribution is made.
 
==Notes==
{{reflist}}
 
== References ==
* Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media.
* Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf]
 
{{ProbDistributions|multivariate}}
 
[[Category:Multivariate continuous distributions]]
[[Category:Conjugate prior distributions]]
[[Category:Normal distribution]]
[[Category:Probability distributions]]

Revision as of 20:42, 10 February 2014

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