Parallel chains

The parallel tempering Markov chain was proposed by Geyer 1991 to help improve mixing by constructing a series of chains that sample from successively broader distributions. Let $f_0$

be the fiducal cold chain. We can construct warmer chains by defining:

$f_j = {1\over C_j} f_0^{1/T_j}$

where $T_0=0$ and $C_j$ is a normalization constant. The sequence of $N$ chains can be ordered such that $1=T_0<T_1<T_2<\cdots<T_N$ . It is straightforward to construct a transition probabilty that preserves detailed balance between adjacent temperature states $j-1$ and $j$ . A new algorithm preserving detailed balance can be enforced by appropriately choosing regular Monte Carlo updates or parallel tempering updates with probabilities that are independent of the configurations of the two systems or of the Monte Carlo step.

The parallel tempering method can acts like a super simulated annealing by tunneling between metastable states and improving convergence to a global optimum without sacrificing the convergence properties of each individual chain.

The related parallel hierarchical sampler is closely related to parallel tempering, using the same sequence of chains. The update step differs in thata swap is proposed between the coldest chain and some other randomly chosen chain at every step, while the remaining $N-2$ chains are updated using the regular (e.g. Metropolis-Hastings) step.

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Documentation generated at Fri Mar 26 00:35:11 2010 by

BIE

Parallel chains