Normal-Wishart distribution
# Normal-Wishart distribution

In probability theory and statistics, the normal-Wishart distribution (or Gaussian-Wishart distribution) is a multivariate four-parameter family of continuous probability distributions. It is the conjugate prior of a multivariate normal distribution with unknown mean and precision matrix (the inverse of the covariance matrix).

## Definition

Suppose

    ${\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Lambda }}\sim {\mathcal {N}}({\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})$

has a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and covariance matrix $(\lambda {\boldsymbol {\Lambda }})^{-1}$, where

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

has a Wishart distribution. Then $({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})$ has a normal-Wishart distribution, denoted as

    $({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu ).$

## Characterization

### Probability density function

    $f({\boldsymbol {\mu }},{\boldsymbol {\Lambda }}|{\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )={\mathcal {N}}({\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})\ {\mathcal {W}}({\boldsymbol {\Lambda }}|\mathbf {W} ,\nu )$

## Properties

### Marginal distributions

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

## Posterior distribution of the parameters

After making $n$ observations ${\boldsymbol {x}}_{1},\dots ,{\boldsymbol {x}}_{n}$, the posterior distribution of the parameters is

    $({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{n},\lambda _{n},\mathbf {W} _{n},\nu _{n}),$

where

    $\lambda _{n}=\lambda +n,$
    ${\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\boldsymbol {\bar {x}}}}{\lambda +n}},$
    $\nu _{n}=\nu +n,$
    $\mathbf {W} _{n}^{-1}=\mathbf {W} ^{-1}+\sum _{i=1}^{n}({\boldsymbol {x}}_{i}-{\boldsymbol {\bar {x}}})({\boldsymbol {x}}_{i}-{\boldsymbol {\bar {x}}})^{T}+{\frac {n\lambda }{n+\lambda }}({\boldsymbol {\bar {x}}}-{\boldsymbol {\mu }}_{0})({\boldsymbol {\bar {x}}}-{\boldsymbol {\mu }}_{0})^{T}.$

## Generating normal-Wishart random variates

Generation of random variates is straightforward:

* Sample ${\boldsymbol {\Lambda }}$ from a Wishart distribution with parameters $\mathbf {W}$ and $\nu$
* Sample ${\boldsymbol {\mu }}$ from a multivariate normal distribution with mean ${\boldsymbol {\mu }}_{0}$ and variance $(\lambda {\boldsymbol {\Lambda }})^{-1}$

## Related distributions

* 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.
