Normal-Wishart distribution

Short description: Multivariate probability distribution

Normal-WishartNotation|  (μ,Λ)∼NW(μ0,λ,𝐖,ν)

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Parameters|  μ0∈ℝD location (vector of real)
λ>0 (real)
𝐖∈ℝD×D scale matrix (pos. def.)
ν>D−1 (real)
Support|  μ∈ℝD;Λ∈ℝD×D covariance matrix (pos. def.)
PDF|  f(μ,Λ|μ0,λ,𝐖,ν)=𝒩(μ|μ0,(λΛ)−1) 𝒲(Λ|𝐖,ν)

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]

## Contents

* 1 Definition
* 2 Characterization
* 2.1 Probability density function
* 3 Properties
* 3.1 Scaling
* 3.2 Marginal distributions
* 4 Posterior distribution of the parameters
* 5 Generating normal-Wishart random variates
* 6 Related distributions
* 7 Notes
* 8 References

## Definition

Suppose

μ|μ0,λ,Λ∼𝒩(μ0,(λΛ)−1)

has a multivariate normal distribution with mean μ0 and covariance matrix (λΛ)−1, where

Λ|𝐖,ν∼𝒲(Λ|𝐖,ν)

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

(μ,Λ)∼NW(μ0,λ,𝐖,ν).

## Characterization

### Probability density function

f(μ,Λ|μ0,λ,𝐖,ν)=𝒩(μ|μ0,(λΛ)−1) 𝒲(Λ|𝐖,ν)

## Properties

### Scaling

### 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.

## Posterior distribution of the parameters

After making n observations x1,…,xn, the posterior distribution of the parameters is

(μ,Λ)∼NW(μn,λn,𝐖n,νn),

where

λn=λ+n,
μn=λμ0+nx¯λ+n,
νn=ν+n,
𝐖n−1=𝐖−1+∑i=1n(xi−x¯)(xi−x¯)T+nλn+λ(x¯−μ0)(x¯−μ0)T.[2]

## Generating normal-Wishart random variates

Generation of random variates is straightforward:

1. Sample Λ from a Wishart distribution with parameters 𝐖 and ν
2. Sample μ from a multivariate normal distribution with mean μ0 and variance (λΛ)−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

1. ↑ Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.
2. ↑ Cross Validated, https://stats.stackexchange.com/q/324925

## References

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

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