While I was reading about Gibbs sampling, I happened to see "simulated annealing" but what is it doing in Gibbs sampling? Although I don't understand the full context of simulated annealing, I am curious about why it is in Gibbs sampling.
Simulated annealing and Gibbs sampling share the same tool of using a Markov chain to explore the surface of a target function, $f$, but the former aims at finding the global maximum while the latter intends to visit the entire surface in proportion to the altitude. Simulated annealing should thus converge to a single point (assuming there is a single maximum) while Gibbs sampling and other MCMC methods keep producing new values that behave like random variables with distribution $f$.
A major theoretical distinction between the two algorithms is that the Markov chain produced by a simulated annealing approach is heterogeneous, that is with a target varying with the iteration $t$, while it is homogeneous for MCMC between both approaches. On the one hand, establishing convergence to the global mode is much harder than establishing that the Markov chain reaches stationarity. On the other hand, since the goal of simulated annealing is a single point, adaptation and other ad hoc manipulations can be included in the algorithm at no higher risk of missing the mode.