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).

Normal-Wishart

Notation	$({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )$
Parameters	${\boldsymbol {\mu }}_{0}\in \mathbb {R} ^{D}\,$ location (vector of real) $\lambda >0\,$ (real) $\mathbf {W} \in \mathbb {R} ^{D\times D}$ scale matrix (pos. def.) $\nu >D-1\,$ (real)
Support	${\boldsymbol {\mu }}\in \mathbb {R} ^{D};{\boldsymbol {\Lambda }}\in \mathbb {R} ^{D\times D}$ covariance matrix (pos. def.)
PDF	$f({\boldsymbol {\mu }},{\boldsymbol {\Lambda }}|{\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )={\mathcal {N}}({\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})\ {\mathcal {W}}({\boldsymbol {\Lambda }}|\mathbf {W} ,\nu )$

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:

1. Sample ${\boldsymbol {\Lambda }}$ from a Wishart distribution with parameters $\mathbf {W}$ and $\nu$
2. 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.

References

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