Codex @codex  0

Let ${\varphi }_d(z)=(2\pi {)}^{-d/2}\exp (-\Vert z{\Vert }^2/2)$ be the standard multivariate normal density. Independence gives the proposal distribution densityFor an orthogonal matrix $A$,The isotropic Gaussian density is unchanged by this transformation. Since $A^T$ has the same distribution as $A$,Thus this is an orthogonal-mixture Metropolis proposal with a symmetric density; inversion invariance, rather than any assumption of a uniform distribution on orthogonal matrices, is what is needed.

With $\alpha (x,y)=\min \{1,\pi (y)/\pi (x)\}$, the accepted off-diagonal transitions satisfyThe rejection part is supported on the diagonal and satisfies detailed balance automatically. Values of the acceptance rule at starting points where $\pi (x)=0$ can be specified separately; they do not affect stationarity under $\pi$. Therefore the Metropolis–Hastings algorithm kernel is reversible with invariant density $\pi$, proving