Gaussian likelihood Gibbs annealing ($({\mu }_n,v_n)⟶(\bar{x},S/m)$)

Let $\bar{x}=m^{-1}{\sum }_i{x}_i$ and $S={\sum }_i(x_i-\bar{x}{)}^2>0$. With Lebesgue base measure $d\mu dv$ for $v={\sigma }^2$, the likelihood-power law has conditionalsThe joint law is proper for $T<m/3$, since its marginal $v$ is inverse-gamma distribution with shape $m/(2T)-3/2$ and scale $S/(2T)$. There is also a direct convergence proof for successive Gibbs sampling updates with deterministic $0<T_n\le m/10$ and $T_n\to 0$. Update ${\mu }_n$ from $v_{n-1}$, then $v_n$ from ${\mu }_n$. Inverse-gamma distribution moments and conditional normal moments giveThe coefficients of the preceding moments are bounded by $1/6$ and $1/8$, so these moments stay bounded from a finite deterministic initial variance. Letting $T_n\to 0$ gives $\mathbb{E}{v}_n\to S/m$ and $\mathbb{E}{v}_n^2\to (S/m{)}^2$. Also $\mathbb{E}({\mu }_n-\bar{x}{)}^2=T_n\mathbb{E}{v}_{n-1}/m\to 0$. Thus the iterates converge in mean square to the normal maximum-likelihood estimate.