|Parameters|| location (vector of real)
scale matrix (pos. def.)
|Support||covariance matrix (pos. def.)|
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).
has a Wishart distribution. Then has a normal-Wishart distribution, denoted as
Probability density function
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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Generating normal-Wishart random variates
Generation of random variates is straightforward:
- Sample from a Wishart distribution with parameters and
- Sample from a multivariate normal distribution with mean and variance
- The normal-inverse Wishart distribution is essentially the same distribution parameterized by variance rather than precision.
- The normal-gamma distribution is the one-dimensional equivalent.
- The multivariate normal distribution and Wishart distribution are the component distributions out of which this distribution is made.
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media. Page 690.
- Bishop, Christopher M. (2006). Pattern Recognition and Machine Learning. Springer Science+Business Media.