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]
Normal-WishartNotation |
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/6b934ecdcbfb1303a5c4979c44543c8455cc4786) |
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Parameters |
location (vector of real)
(real)
scale matrix (pos. def.)
(real) |
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Support |
covariance matrix (pos. def.) |
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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 )}](http://178.128.105.246/cars-https-wikimedia.org/api/rest_v1/media/math/render/svg/ee18e740872ad02698aa9effa54e6d270c3bb65e) |
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Suppose
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has a multivariate normal distribution with mean and covariance matrix , where
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has a Wishart distribution. Then
has a normal-Wishart distribution, denoted as
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Probability density function
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Posterior distribution of the parameters
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After making observations , the posterior distribution of the parameters is
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where
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- [2]
Generating normal-Wishart random variates
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Generation of random variates is straightforward:
- Sample from a Wishart distribution with parameters and
- Sample from a multivariate normal distribution with mean and variance
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.