normal-Wishart distribution

Suppose

:$\backslash boldsymbol\backslash mu|\backslash boldsymbol\backslash mu\_0,\backslash lambda,\backslash boldsymbol\backslash Lambda\; \backslash sim\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu\_0,(\backslash lambda\backslash boldsymbol\backslash Lambda)^\{-1\})$

has a multivariate normal distribution with mean $\backslash boldsymbol\backslash mu\_0$ and covariance matrix $(\backslash lambda\backslash boldsymbol\backslash Lambda)^\{-1\}$, where

:$\backslash boldsymbol\backslash Lambda|\backslash mathbf\{W\},\backslash nu\; \backslash sim\; \backslash mathcal\{W\}(\backslash boldsymbol\backslash Lambda|\backslash mathbf\{W\},\backslash nu)$

has a Wishart distribution. Then $(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Lambda)$

has a normal-Wishart distribution, denoted as

:$(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Lambda)\; \backslash sim\; \backslash mathrm\{NW\}(\backslash boldsymbol\backslash mu\_0,\backslash lambda,\backslash mathbf\{W\},\backslash nu)\; .$

: $f(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Lambda|\backslash boldsymbol\backslash mu\_0,\backslash lambda,\backslash mathbf\{W\},\backslash nu)\; =\; \backslash mathcal\{N\}(\backslash boldsymbol\backslash mu|\backslash boldsymbol\backslash mu\_0,(\backslash lambda\backslash boldsymbol\backslash Lambda)^\{-1\})\backslash \; \backslash mathcal\{W\}(\backslash boldsymbol\backslash Lambda|\backslash mathbf\{W\},\backslash nu)$

By construction, the marginal distribution over $\backslash boldsymbol\backslash Lambda$ is a Wishart distribution, and the conditional distribution over $\backslash boldsymbol\backslash mu$ given $\backslash boldsymbol\backslash Lambda$ is a multivariate normal distribution. The marginal distribution over $\backslash boldsymbol\backslash mu$ is a multivariate t-distribution.

After making $n$ observations $\backslash boldsymbol\{x\}\_1,\; \backslash dots,\; \backslash boldsymbol\{x\}\_n$, the posterior distribution of the parameters is

:$(\backslash boldsymbol\backslash mu,\backslash boldsymbol\backslash Lambda)\; \backslash sim\; \backslash mathrm\{NW\}(\backslash boldsymbol\backslash mu\_n,\backslash lambda\_n,\backslash mathbf\{W\}\_n,\backslash nu\_n),$

where

:$\backslash lambda\_n\; =\; \backslash lambda\; +\; n,$

:$\backslash boldsymbol\backslash mu\_n\; =\; \backslash frac\{\backslash lambda\; \backslash boldsymbol\backslash mu\_0\; +\; n\backslash boldsymbol\{\backslash bar\{x\}\}\}\{\backslash lambda\; +\; n\},$

:$\backslash nu\_n\; =\; \backslash nu\; +\; n,$

:$\backslash mathbf\{W\}\_n^\{-1\}\; =\; \backslash mathbf\{W\}^\{-1\}\; +\; \backslash sum\_\{i=1\}^n\; (\backslash boldsymbol\{x\}\_i\; -\; \backslash boldsymbol\{\backslash bar\{x\}\})(\backslash boldsymbol\{x\}\_i\; -\; \backslash boldsymbol\{\backslash bar\{x\}\})^T\; +\; \backslash frac\{n\; \backslash lambda\}\{n\; +\; \backslash lambda\}\; (\backslash boldsymbol\{\backslash bar\{x\}\}\; -\; \backslash boldsymbol\backslash mu\_0)(\backslash boldsymbol\{\backslash bar\{x\}\}\; -\; \backslash boldsymbol\backslash mu\_0)^T.$Cross Validated, https://stats.stackexchange.com/q/324925

Generation of random variates is straightforward:

1. Sample $\backslash boldsymbol\backslash Lambda$ from a Wishart distribution with parameters $\backslash mathbf\{W\}$ and $\backslash nu$
2. Sample $\backslash boldsymbol\backslash mu$ from a multivariate normal distribution with mean $\backslash boldsymbol\backslash mu\_0$ and variance $(\backslash lambda\backslash boldsymbol\backslash Lambda)^\{-1\}$

• The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
• The normal-gamma distribution is the one-dimensional equivalent.
• The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.

{{reflist}}

• Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.

{{ProbDistributions|multivariate}}

{{DEFAULTSORT:Normal-Wishart Distribution}}

Category:Multivariate continuous distributions

Category:Conjugate prior distributions

Category:Normal distribution