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Revision as of 04:57, 27 April 2016 by 130.102.82.62(talk)(→Marginal distributions: there was a mistake not consistent with the Koch's book. The variance-covariance matrix of the Student distribution must be V^{-1}.)
In a multivariate form of the normal-inverse-gamma distribution, -- that is, conditional on , is a random vector that follows the multivariate normal distribution with mean and covariance -- while, as in the univariate case, .
Characterization
Probability density function
For the multivariate form where is a random vector,
where is the determinant of the matrix. Note how this last equation reduces to the first form if so that are scalars.
Alternative parameterization
It is also possible to let in which case the pdf becomes
In the multivariate form, the corresponding change would be to regard the covariance matrix instead of its inverse as a parameter.
Cumulative distribution function
Differential equation
The probability density function of the normal-inverse-gamma distribution is a solution to the following differential equation:
A generalization of this distribution which allows for a multivariate mean and a completely unknown positive-definite covariance matrix (whereas in the multivariate inverse-gamma distribution the covariance matrix is regarded as known up to the scale factor ) is the normal-inverse-Wishart distribution
References
Denison, David G. T. ; Holmes, Christopher C.; Mallick, Bani K.; Smith, Adrian F. M. (2002) Bayesian Methods for Nonlinear Classification and Regression, Wiley. ISBN 0471490369
Koch, Karl-Rudolf (2007) Introduction to Bayesian Statistics (2nd Edition), Springer. ISBN 354072723X