= Likelihood-power annealing
{title2=$\pi_T(\theta)\propto L(\theta)^{1/T}$}
To maximize a <log-likelihood>, use energy $-\ell$ and <Boltzmann distribution> proportional to $e^{\ell/T}=L^{1/T}$ at temperature $T>0$. Reversing this sign favors <likelihood> minima and can make the <probability density function> nonnormalizable. For Poisson observations with total $s$ and sample size $m$, with respect to $d\lambda$ the normalized <probability density function> is
$$
\lambda\sim\operatorname{Gamma}(s/T+1,m/T).
$$
Its <mean> is $(s+T)/m$ and <variance> $(sT+T^2)/m^2$, so it concentrates at the <maximum-likelihood estimate> $s/m$ as $T\downarrow0$. Cooling of equilibrium laws and convergence of a particular changing-temperature chain are different statements; the latter requires mixing or a direct chain argument.
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