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Template:Probability 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

$𝝁|{𝝁}_0,\lambda ,𝜦\sim 𝒩(𝝁|{𝝁}_0,(\lambda 𝜦{)}^{-1})$

has a multivariate normal distribution with mean ${𝝁}_0$ and covariance matrix $(\lambda 𝜦{)}^{-1}$, where

$𝜦|𝐖,\nu \sim 𝒲(𝜦|𝐖,\nu )$

has a Wishart distribution. Then $(𝝁,𝜦)$ has a normal-Wishart distribution, denoted as

$(𝝁,𝜦)\sim \mathrm{N}\mathrm{W}({𝝁}_0,\lambda ,𝐖,\nu ).$

Characterization

Probability density function

$f(𝝁,𝜦|{𝝁}_0,\lambda ,𝐖,\nu )=𝒩(𝝁|{𝝁}_0,(\lambda 𝜦{)}^{-1})𝒲(𝜦|𝐖,\nu )$

Properties

Marginal distributions

By construction, the marginal distribution over $𝜦$ is a Wishart distribution, and the conditional distribution over $𝝁$ given $𝜦$ is a multivariate normal distribution. The marginal distribution over $𝝁$ is a multivariate t-distribution.

Generating normal-Wishart random variates

Generation of random variates is straightforward:

1. Sample $𝜦$ from a Wishart distribution with parameters $𝐖$ and $\nu$
2. Sample $𝝁$ from a multivariate normal distribution with mean ${𝝁}_0$ and variance $(\lambda 𝜦{)}^{-1}$

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.

Notes

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References

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

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