Notation | |||
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Parameters |
location (vector of real) (real) inverse scale matrix (pos. def.) (real) | ||
Support | covariance matrix (pos. def.) | ||
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[edit]
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[edit]
Probability density function[edit]
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[edit]
Scaling[edit]
Marginal distributions[edit]
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[edit]
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]
Generating normal-inverse-Wishart random variates[edit]
Generation of random variates is straightforward:
- Sample from an inverse Wishart distribution with parameters and
- Sample from a multivariate normal distribution with mean and variance
Related distributions[edit]
- 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[edit]
- ^ 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[edit]
- 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]