🚀 go-pugleaf

In article <92Oct5.123636edt.472@neuron.ai.toronto.edu> radford@cs.toronto.edu
(Radford Neal) writes:

> The technique consists (in its simplest form) of splitting the
> simulation up into several phases, each associated with a certain
> "temperature", T. These temperatures decline, starting at a high
> value, and ending at T=1. In each phase, a Markov chain that has an
> equilibrium distribution proportional to P(x)^(1/T) is simulated,
> where P(x) is the distribution we wish to sample from. The start state
> for each phase is the final state from the previous phase.
>
> Once a temperature of one is reached, or perhaps a bit later, one
> starts using the states visited by this final chain (with T=1) to
> calculate expectations with respect to the distribution given by
> P(x). 

But this doesn't help much.  All it attempts to do is produce one
realization that is "almost" from the stationary distribution to
use as the start of the run.

But if the chain mixes very slowly, the average will be near the starting
point, but not near the expectation one wants to calculate.

On the other hand, if the chain mixes rapidly, the "annealing" isn't
needed.

A rather different algorithm, in which the chain is slowly "annealed"
and its final state is treated as a single realization from the stationary
distribution (and one needs a separate annealing run for each realization
simulated) makes some sense, but is horribly slow and (as I said) throws
Markov chain theory out the window.

-- 
Charles Geyer
School of Statistics
University of Minnesota
charlie@umnstat.stat.umn.edu