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Normal-WishartNotation |
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu )}](https://wikimedia.riteme.site/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 )}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ee18e740872ad02698aa9effa54e6d270c3bb65e) |
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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]
Suppose
![{\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Lambda }}\sim {\mathcal {N}}({\boldsymbol {\mu }}_{0},(\lambda {\boldsymbol {\Lambda }})^{-1})}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/6d2cc90890e646274373a48831e5a34050704536)
has a multivariate normal distribution with mean
and covariance matrix
, where
![{\displaystyle {\boldsymbol {\Lambda }}|\mathbf {W} ,\nu \sim {\mathcal {W}}({\boldsymbol {\Lambda }}|\mathbf {W} ,\nu )}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/b1ec50704216114758639181fd3622f8f3d167f6)
has a Wishart distribution. Then
has a normal-Wishart distribution, denoted as
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Lambda }})\sim \mathrm {NW} ({\boldsymbol {\mu }}_{0},\lambda ,\mathbf {W} ,\nu ).}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/d41bd515b4ac2316468a10b0fe9a8d00a259e57d)
Probability density function
[edit]
![{\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 )}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/ee18e740872ad02698aa9effa54e6d270c3bb65e)
Marginal distributions
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By construction, the marginal distribution over
is a Wishart distribution, and the conditional distribution over
given
is a multivariate normal distribution. The marginal distribution over
is a multivariate t-distribution.
Posterior distribution of the parameters
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After making
observations
, 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}),}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/1f783a11ed298e91a6be7b1165b64593d4090dd6)
where
![{\displaystyle \lambda _{n}=\lambda +n,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/31bfb2bad6a64dff57ad58381e247ae521ca5b84)
![{\displaystyle {\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\boldsymbol {\bar {x}}}}{\lambda +n}},}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/deecd7f55346536467dc46290484c9642fcebe47)
![{\displaystyle \nu _{n}=\nu +n,}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/69e54e2218b4d3c8ba00031f332764bee6647935)
[2]
Generating normal-Wishart random variates
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Generation of random variates is straightforward:
- Sample
from a Wishart distribution with parameters
and ![{\displaystyle \nu }](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
- Sample
from a multivariate normal distribution with mean
and variance ![{\displaystyle (\lambda {\boldsymbol {\Lambda }})^{-1}}](https://wikimedia.riteme.site/api/rest_v1/media/math/render/svg/0610445dd263f1912dc1e6ce6a561e3810ffcc4b)
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
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Discrete univariate | with finite support | |
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with infinite support | |
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Continuous univariate | supported on a bounded interval | |
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supported on a semi-infinite interval | |
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supported on the whole real line | |
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with support whose type varies | |
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Mixed univariate | |
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Multivariate (joint) | |
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Directional | |
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Degenerate and singular | |
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Families | |
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