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Inverse gamma distribution is a special case of type 5 Pearson distribution; ... , and recall that the pdf of the gamma distribution is = (), >. Note that ...
If the shape parameter of the gamma distribution is known, but the inverse-scale parameter is unknown, then a gamma distribution for the inverse scale forms a conjugate prior. The compound distribution, which results from integrating out the inverse scale, has a closed-form solution known as the compound gamma distribution. [22]
In probability theory and statistics, the normal-inverse-gamma distribution (or Gaussian-inverse-gamma distribution) is a four-parameter family of multivariate continuous probability distributions. It is the conjugate prior of a normal distribution with unknown mean and variance .
i.e., the inverse-gamma distribution, where () is the ordinary Gamma function. The Inverse Wishart distribution is a special case of the inverse matrix gamma distribution when the shape parameter = and the scale parameter =. Another generalization has been termed the generalized inverse Wishart distribution, .
where (;,) is the gamma pdf with shape and inverse scale . The mode, mean and variance of the compound gamma can be obtained by multiplying the mode and mean in the above infobox by q and the variance by q 2.
The Wishart distribution is related to the inverse-Wishart distribution, denoted by , as follows: If X ~ W p (V, n) and if we do the change of variables C = X −1, then (,). This relationship may be derived by noting that the absolute value of the Jacobian determinant of this change of variables is | C | p +1 , see for example equation (15.15 ...
Inverse distributions arise in particular in the Bayesian context of prior distributions and posterior distributions for scale parameters. In the algebra of random variables , inverse distributions are special cases of the class of ratio distributions , in which the numerator random variable has a degenerate distribution .
which is quite close to the complex multivariate pdf of G itself. The elements of G conventionally have circular symmetry such that E [ G G T ] = 0 {\displaystyle \mathbb {E} [GG^{T}]=0} . Inverse Complex Wishart The distribution of the inverse complex Wishart distribution of Y = S − 1 {\displaystyle \mathbf {Y} =\mathbf {S^{-1}} } according ...