= Solution
At temperature $T>0$, target a <probability density function> proportional to $\exp(f(\theta)/T)$ on a parameter region where it is normalizable, possibly with a specified proper base measure. A <Metropolis–Hastings algorithm> proposal $q(\theta,\theta')$ has acceptance <probability>
$$
\min\left\{1,\exp\left[\frac{f(\theta')-f(\theta)}T\right]\frac{q(\theta',\theta)}{q(\theta,\theta')}\right\}.
$$
For a symmetric proposal, improving moves are accepted and a loss $\Delta$ is accepted with <probability> $e^{-\Delta/T}$. Run updates while gradually lowering $T$ toward zero. This <simulated annealing> scheme initially permits exploration and increasingly favors high objective values. Appropriate cooling and mixing conditions are needed for a general global convergence theorem; a finite arbitrary schedule does not guarantee finding a global maximum.
To maximize a positive <likelihood function>, take $f=\log L$, which has the same maximizers. The resulting <likelihood-power annealing> target is $L^{1/T}$, and available <Gibbs sampler> <conditional distributions> can replace Metropolis proposals.
Back to article page
