Normal-Wishart

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

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).^[1]

Definition

Suppose

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

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

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

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

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

Characterization

Probability density function

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

Properties

Scaling

Marginal distributions

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

Posterior distribution of the parameters

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

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

where

${\displaystyle \lambda _{n}=\lambda +n,}$

${\displaystyle {\boldsymbol {\mu ))_{n}={\frac {\lambda {\boldsymbol {\mu ))_{0}+n{\boldsymbol {\bar {x)))){\lambda +n)),}$

${\displaystyle \nu _{n}=\nu +n,}$

${\displaystyle \mathbf {W} _{n}^{-1}=\mathbf {W} ^{-1}+\sum _{i=1}^{n}({\boldsymbol {x))_{i}-{\boldsymbol {\bar {x))})({\boldsymbol {x))_{i}-{\boldsymbol {\bar {x))})^{T}+{\frac {n\lambda }{n+\lambda ))({\boldsymbol {\bar {x))}-{\boldsymbol {\mu ))_{0})({\boldsymbol {\bar {x))}-{\boldsymbol {\mu ))_{0})^{T}.}$^[2]

Generating normal-Wishart random variates

Generation of random variates is straightforward:

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

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.