Multivariate inverse-gamma distribution with independent components.
This is typically used to establish a conjugate prior for a Bayesian multivariate linear regression with number of outputs that are conditionally independent given the inputs:
where subscript denotes the (hyper)parameters of the th element of the output vector, are inputs, and are outputs.
The relationship is established in code as follows:
σ2:Random<Real[_]>; α:Real; β:Real[_]; W:Random<Real[_,_]>; M:Real[_,_]; U:Real[_,_]; Y:Random<Real[_,_]>; X:Real[_,_]; σ2 ~ InverseGamma(α, β); W ~ Gaussian(M, U, σ2); Y ~ Gaussian(W*X, σ2);
The advantage of using this approach over separate regressions is that expensive covariance operations are shared.