 # ⓘ Normal-Wishart distribution. In probability theory and statistics, the normal-Wishart distribution is a multivariate four-parameter family of continuous probabi ..

## ⓘ Normal-Wishart distribution

In probability theory and statistics, the normal-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.

## 1. Definition

Suppose

μ | μ 0, λ, Λ ∼ N μ 0, Λ − 1) {\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 μ 0 {\displaystyle {\boldsymbol {\mu }}_{0}} and covariance matrix Λ − 1 {\displaystyle \lambda {\boldsymbol {\Lambda }}^{-1}}, where

Λ | W, ν ∼ W Λ | W, ν {\displaystyle {\boldsymbol {\Lambda }}|\mathbf {W},\nu \sim {\mathcal {W}}{\boldsymbol {\Lambda }}|\mathbf {W},\nu}

has a Wishart distribution. Then μ, Λ {\displaystyle {\boldsymbol {\mu }},{\boldsymbol {\Lambda }}} has a normal-Wishart distribution, denoted as

μ, Λ ∼ N W μ 0, λ, W, ν. {\displaystyle {\boldsymbol {\mu }},{\boldsymbol {\Lambda }}\sim \mathrm {NW} {\boldsymbol {\mu }}_{0},\lambda,\mathbf {W},\nu.}

## 2. Characterization

### Probability density function

f = N μ | μ 0, Λ − 1) W Λ | W, ν {\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}

## 3. Properties

### Marginal distributions

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

## 4. Posterior distribution of the parameters

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

μ, Λ ∼ N W, {\displaystyle {\boldsymbol {\mu }},{\boldsymbol {\Lambda }}\sim \mathrm {NW} {\boldsymbol {\mu }}_{n},\lambda _{n},\mathbf {W} _{n},\nu _{n},}

where

λ n = λ + n, {\displaystyle \lambda _{n}=\lambda +n,} μ n = λ μ 0 + n x ¯ λ + n, {\displaystyle {\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\boldsymbol {\bar {x}}}}{\lambda +n}},} ν n = ν + n, {\displaystyle \nu _{n}=\nu +n,} W n − 1 = W − 1 + ∑ i = 1 n x i − x ¯ x i − x ¯ T + n λ n + λ x ¯ − μ 0 x ¯ − μ 0 T. {\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}.}

## 5. Generating normal-Wishart random variates

Generation of random variates is straightforward:

• Sample Λ {\displaystyle {\boldsymbol {\Lambda }}} from a Wishart distribution with parameters W {\displaystyle \mathbf {W} } and ν {\displaystyle \nu }
• Sample μ {\displaystyle {\boldsymbol {\mu }}} from a multivariate normal distribution with mean μ 0 {\displaystyle {\boldsymbol {\mu }}_{0}} and variance Λ − 1 {\displaystyle \lambda {\boldsymbol {\Lambda }}^{-1}}

## 6. Related distributions

• The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.
• 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.

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