Issue 06 — `InverseWishart` default-space log-partition uses `mvtrigamma` instead of `logmvgamma`

[docxology]

Issue 06 — InverseWishart default-space log-partition uses mvtrigamma instead of logmvgamma

Component: src/distributions/wishart_inverse.jl:323-327
Severity: MEDIUM (silent wrong value; mean-space only)
Type: Correctness

Summary

The DefaultParametersSpace (mean parameter space) log-partition of
InverseWishartFast is wrong:

getlogpartition(::DefaultParametersSpace, ::Type{InverseWishartFast}) = (θ) -> begin
    (ν, S) = unpack_parameters(InverseWishartFast, θ)
    p = first(size(S))
    return (ν / 2) * (p * log(2.0) - logdet(S)) + mvtrigamma(p, ν / 2)   # ← mvtrigamma!
end

mvtrigamma is the sum of trigammas — the correct object here is the log of the
multivariate gamma function, logmvgamma(p, ν/2). The Wishart sibling
(src/distributions/wishart.jl:297) uses logmvgamma correctly for the analogous
term. This looks like a copy-paste of the Fisher [1,1] entry (which legitimately
uses mvtrigamma).

Evidence / repro (Julia 1.12.6)

ν = 6.0; S = [3.0 0.5; 0.5 2.0]; p = 2
A_default = getlogpartition(DefaultParametersSpace(), InverseWishartFast)(θ)   # -0.2034
A_natural = getlogpartition(NaturalParametersSpace(), InverseWishartFast)(η)   #  0.4615
# hand value with logmvgamma: (ν/2)(p·ln2 − logdet S) + logmvgamma(p,ν/2) = 0.4615

The two parameter spaces return different values for the same log-partition; the
natural space and hand-derived values agree at 0.4615.

Impact

The mean-space log-partition of InverseWishartFast (used in ReactiveMP
NodeFunctionRule-type expectations and sqrt/mean-space computations) silently
returns a wrong value. Mathematically the two parametrizations must give the same
partition.

Proposed fix

return (ν / 2) * (p * log(2.0) - logdet(S)) + logmvgamma(p, ν / 2)