Sampling from degenerate multivariate normal.

[mschauer]

Including a degenerate multivariate normal would be nice. Some concepts, e.g. gradlogpdf(d, x) would need a different interpretation and could be skipped for now.

[lindahua]

This means zero variance along certain directions. It is interpretable mathematically.

However, it is tricky in implementation. As much of the computation relies on Cholesky decomposition, which requires positive definiteness.

[mschauer]

The random variable Lz + μ where L is a dxℓ-matrix and LL' = A and z ~ Nℓ(0, I) is well defined and said to have a multivariate normal distribution Nd(μ, A). It is not absolutely continuous on R^d, but has density with respect to the Lebesgue measure on an adequate subspace, see http://en.wikipedia.org/wiki/Multivariate_normal_distribution#Degenerate_case . Given μ and A sampling is possible for example via

function randsmvn(mu, A) 
    S = cholfact(A, :L, Val{true})
    return mu + ipermute!(S[:L]*permute!(randn(size(S,1)),S[:p]), S[:p]) 
end

[lindahua]

I think the issue should be to PDMats.jl, to add a degenerated covariance matrix type there.

[mschauer]

Ref JuliaStats/PDMats.jl#25