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| {{Probability distribution |
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| name =normal-inverse-Wishart|
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| type =density|
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| pdf_image =|
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| cdf_image =|
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| notation =<math>(\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathrm{NIW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu)</math>|
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| 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)|
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| support =<math>\boldsymbol\mu\in\mathbb{R}^D ; \boldsymbol\Sigma \in\mathbb{R}^{D\times D}</math> [[covariance matrix]] ([[positive-definite matrix|pos. def.]])|
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| 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>|
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| cdf =|
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| mean =|
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| median =|
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| mode =|
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| variance =|
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| skewness =|
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| kurtosis =|
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| entropy =|
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| char =|
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| }}
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| 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>
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| ==Definition==
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| Suppose
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| :<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>
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| has a [[multivariate normal distribution]] with [[mean]] <math>\boldsymbol\mu_0</math> and [[covariance matrix]] <math>\tfrac{1}{\lambda}\boldsymbol\Sigma</math>, where
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| :<math>\boldsymbol\Sigma|\boldsymbol\Psi,\nu \sim \mathcal{W}^{-1}(\boldsymbol\Sigma|\boldsymbol\Psi,\nu)</math>
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| has an [[inverse Wishart distribution]]. Then <math>(\boldsymbol\mu,\boldsymbol\Sigma) </math>
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| has a normal-inverse-Wishart distribution, denoted as
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| :<math> (\boldsymbol\mu,\boldsymbol\Sigma) \sim \mathrm{NIW}(\boldsymbol\mu_0,\lambda,\boldsymbol\Psi,\nu) .
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| </math>
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| ==Characterization==
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| ===Probability density function===
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| : <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>
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| ==Properties==
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| ===Scaling===
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| ===Marginal distributions===
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| 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]].
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| == Posterior distribution of the parameters ==
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| {{Empty section|date=March 2013}}
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| == Generating normal-inverse-Wishart random variates ==
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| Generation of random variates is straightforward:
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| # Sample <math>\boldsymbol\Sigma</math> from an [[inverse Wishart distribution]] with parameters <math>\boldsymbol\Psi</math> and <math>\nu</math>
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| # 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>
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| == Related distributions ==
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| * 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> .
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| * The [[normal-inverse-gamma distribution]] is the one-dimensional equivalent.
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| * The [[multivariate normal distribution]] and [[inverse Wishart distribution]] are the component distributions out of which this distribution is made.
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| ==Notes==
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| {{reflist}}
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| == References ==
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| * Bishop, Christopher M. (2006). ''Pattern Recognition and Machine Learning.'' Springer Science+Business Media.
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| * Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [http://www.cs.ubc.ca/~murphyk/Papers/bayesGauss.pdf]
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| {{ProbDistributions|multivariate}}
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| [[Category:Multivariate continuous distributions]]
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| [[Category:Conjugate prior distributions]]
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| [[Category:Normal distribution]]
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| [[Category:Probability distributions]]
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Nurse Educator Toney from Holland Landing, enjoys to spend time saltwater aquariums, ganhando dinheiro na internet and wood working. Finds travel an amazing experience after making a trip to Ancient City of Damascus.
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