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For a (possibly signed) nondegenerate probability measure $\pi$ on $\{1,\dots,n\}$ define $$\langle \pi \rangle := \{R \in \operatorname{STO}(n): \pi R = \pi \}.$$ Here $\operatorname{STO}(n)$ 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 $\langle R \rangle \equiv \pi$ for $R \in \langle \pi \rangle$. $\langle \pi \rangle$ is a $(n-1)^2$-dimensional Lie group under matrix multiplication. Its Lie algebra $\mathfrak{lie}(\langle \pi \rangle)$ has a basis of the form $$e_{(j,k)}^{\langle \pi \rangle} := e_{(j,k)} - \frac{\pi_j}{\pi_n}e_{(n,k)}, \quad 1 \le j, k \le n-1$$ where $e_{(j,k)} = e_j(e_k^* - e_n^*)$ and we use typical notation for the standard basis of $\mathbb{R}^n$. A block matrix decomposition shows that $\mathfrak{lie}(\langle \pi \rangle) \cong \mathfrak{gl}(n-1)$, and it is easy to show that $$\exp Ce_{(j,k)}^{\langle \pi \rangle} = I + \frac{e^{C(\pi_j/\pi_n + \delta_{jk})} - 1}{\pi_j/\pi_n + \delta_{jk}} e_{(j,k)}^{\langle \pi \rangle}.$$ 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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  • $\begingroup$ I'm probably missing something, but why is an element of STO(n) necessarily invertible? i.e. why do we have a group and not a semigroup? Indeed, if we take $\pi$ to be the uniform distribution, then doesn't $\langle \pi\rangle$ contain a matrix where all entries have the same value? $\endgroup$ – Yemon Choi Feb 9 '10 at 2:35
  • $\begingroup$ (also, perhaps double dollar rather than single to make the 2nd formula a little easier on the eyesight?) $\endgroup$ – Yemon Choi Feb 9 '10 at 2:46
  • $\begingroup$ I fixed up the LaTeX a bit to make it more readable. Also, some backquotes fixed it so the curly brackets {…} are visible. $\endgroup$ – Harald Hanche-Olsen Feb 9 '10 at 3:44
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    $\begingroup$ I should have said that STO(n) is the set of nonsingular stochastic matrices. Editing to fix. It's isomorphic to the affine group. See, e.g., jstor.org/pss/2974507 $\endgroup$ – Steve Huntsman Feb 9 '10 at 13:21
  • $\begingroup$ Your second equation reminds me of detailed balance but I may be completely off. $\endgroup$ – lcv Dec 11 '18 at 22:08

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