Multivariate parameter family of continuous probability distributions
normal-inverse-WishartNotation | ![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98c2e4e8dafc141669af0ac2adadec0ec1cfe352) |
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Parameters | location (vector of real)
(real)
inverse scale matrix (pos. def.)
(real) |
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Support | covariance matrix (pos. def.) |
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PDF | ![{\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\mathcal {N}}({\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},{\tfrac {1}{\lambda }}{\boldsymbol {\Sigma }})\ {\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2842dad1aac931b2641083e47e3cd2cd52db75ed) |
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In probability theory and statistics, the normal-inverse-Wishart distribution (or Gaussian-inverse-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 covariance matrix (the inverse of the precision matrix).[1]
Definition
Suppose
![{\displaystyle {\boldsymbol {\mu }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Sigma }}\sim {\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/437ef1668b00322c3ca4d5c6feb0117cfd8da5e8)
has a multivariate normal distribution with mean
and covariance matrix
, where
![{\displaystyle {\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu \sim {\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a519bca315507bf03617d8f6ca02ba33a7a7faea)
has an inverse Wishart distribution. Then
has a normal-inverse-Wishart distribution, denoted as
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d264df238ef2f89eecd6415df710bb1e1115cfe4)
Characterization
Probability density function
![{\displaystyle f({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|{\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu )={\mathcal {N}}\left({\boldsymbol {\mu }}{\Big |}{\boldsymbol {\mu }}_{0},{\frac {1}{\lambda }}{\boldsymbol {\Sigma }}\right){\mathcal {W}}^{-1}({\boldsymbol {\Sigma }}|{\boldsymbol {\Psi }},\nu )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6c5b86f41bc9b6292f7ffaa2b58ccca1e1d54675)
The full version of the PDF is as follows:[2]
Here
is the multivariate gamma function and
is the Trace of the given matrix.
Properties
Scaling
Marginal distributions
By construction, the marginal distribution over
is an inverse 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
Suppose the sampling density is a multivariate normal distribution
![{\displaystyle {\boldsymbol {y_{i}}}|{\boldsymbol {\mu }},{\boldsymbol {\Sigma }}\sim {\mathcal {N}}_{p}({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e63f9cd3eb014477280cb78e99de6610980ff0f1)
where
is an
matrix and
(of length
) is row
of the matrix .
With the mean and covariance matrix of the sampling distribution is unknown, we can place a Normal-Inverse-Wishart prior on the mean and covariance parameters jointly
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }})\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{0},\lambda ,{\boldsymbol {\Psi }},\nu ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d264df238ef2f89eecd6415df710bb1e1115cfe4)
The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart
![{\displaystyle ({\boldsymbol {\mu }},{\boldsymbol {\Sigma }}|y)\sim \mathrm {NIW} ({\boldsymbol {\mu }}_{n},\lambda _{n},{\boldsymbol {\Psi }}_{n},\nu _{n}),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b164adca832d80b5522de5198e6ea8227097b53)
where
![{\displaystyle {\boldsymbol {\mu }}_{n}={\frac {\lambda {\boldsymbol {\mu }}_{0}+n{\bar {\boldsymbol {y}}}}{\lambda +n}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64cc80174f172aa439a204fa544e49ef6bcba190)
![{\displaystyle \lambda _{n}=\lambda +n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/647f47932a28c964898103c3992485127db369ce)
![{\displaystyle \nu _{n}=\nu +n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e3be3a7b633fdd1cbbfd10bfe3d8da9dbfcbfd0b)
.
To sample from the joint posterior of
, one simply draws samples from
, then draw
. To draw from the posterior predictive of a new observation, draw
, given the already drawn values of
and
.[3]
Generating normal-inverse-Wishart random variates
Generation of random variates is straightforward:
- Sample
from an inverse Wishart distribution with parameters
and ![{\displaystyle \nu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c15bbbb971240cf328aba572178f091684585468)
- Sample
from a multivariate normal distribution with mean
and variance ![{\displaystyle {\boldsymbol {\tfrac {1}{\lambda }}}{\boldsymbol {\Sigma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/99771ec77b0e8b45449f81b794af769a5f3020be)
Related distributions
- The normal-Wishart distribution is essentially the same distribution parameterized by precision rather than variance. If
then
. - The normal-inverse-gamma distribution is the one-dimensional equivalent.
- The multivariate normal distribution and inverse Wishart distribution are the component distributions out of which this distribution is made.
Notes
- ^ Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [1]
- ^ Simon J.D. Prince(June 2012). Computer Vision: Models, Learning, and Inference. Cambridge University Press. 3.8: "Normal inverse Wishart distribution".
- ^ Gelman, Andrew, et al. Bayesian data analysis. Vol. 2, p.73. Boca Raton, FL, USA: Chapman & Hall/CRC, 2014.
References
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.
- Murphy, Kevin P. (2007). "Conjugate Bayesian analysis of the Gaussian distribution." [2]
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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Category Commons |