Multivariate parameter family of continuous probability distributions
normal-inverse-WishartNotation |
 |
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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.) |
---|
PDF |
 |
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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

has a multivariate normal distribution with mean
and covariance matrix
, where

has an inverse Wishart distribution. Then
has a normal-inverse-Wishart distribution, denoted as

Characterization
Probability density function

The full version of the PDF is as follows:[2]
Here
is the multivariate gamma function and
is the Trace of the given matrix.
Posterior distribution of the parameters
Suppose the sampling density is a multivariate normal distribution

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

The resulting posterior distribution for the mean and covariance matrix will also be a Normal-Inverse-Wishart

where



.
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]