For a (possibly signed) nondegenerate probability measure pi on 1,dots,n define
langlepirangle:=RinoperatornameSTO(n):piR=pi.
Here operatornameSTO(n) denotes the group of invertible stochastic matrices, i.e., the set of matrices (with entries of either sign) whose row sums are unity. Also, write langleRrangleequivpi for Rinlanglepirangle. langlepirangle is a (n−1)2-dimensional Lie group under matrix multiplication. Its Lie algebra mathfraklie(langlepirangle) has a basis of the form
elanglepirangle(j,k):=e(j,k)−fracpijpine(n,k),quad1lej,klen−1
where e(j,k)=ej(e∗k−e∗n) and we use typical notation for the standard basis of mathbbRn. A block matrix decomposition shows that mathfraklie(langlepirangle)congmathfrakgl(n−1), and it is easy to show that
expCelanglepirangle(j,k)=I+fraceC(pij/pin+deltajk)−1pij/pin+deltajkelanglepirangle(j,k).
In particular, this object seems pretty tractable (although I haven't bothered to think about the Haar measure).
My question is, has it been used in
Bayesian statistics (or elsewhere)? It seems
naturally suited for such an
application.
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