# Metropolis-coupled Markov chain Monte Carlo

**Metropolis-coupled Markov chain Monte Carlo** (or **MCMCMC** or **(MC) ^{3}**) is a variant of Markov chain Monte Carlo in which multiple chains are run in parallel, each with a different "temperature" (Geyer, 1991). Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.

"Some of these chains are 'heated' by raising the posterior probability to a power β. For example, if

f(ψ|) is the posterior probability density distribution of the phylogenetic parameters, then a heated version of the posterior distribution isXf(ψ|)X^{β}. Here, β(0 < β < 1) is the heat value of the chain. Heating a Markov chain increases the acceptance probability of new states. A heated chain tends to accept more states than a cold chain, allowing a heated chain to more readily cross valleys in the landscapes of trees." (Altekaret al., 2004).

(MC)^{3} can be used to empirically determine the posterior probability distribution of trees, branch lengths, and substitution parameters.

- Temperature
- A variable used in Metropolis-coupled Markov chain Monte Carlo. The temperature affects the likelihood of acceptance of a proposed tree and also affects the likelihood that two chains will accept a proposed tree swap.

## Practice

- Only one chain is sampled
- The other chains are heated (i.e. they can take bigger steps)
- Chains can swap states
- Allows crossing of valleys
- This process of linking heated chains is called "Metropolis-coupling"
- The full-fledged Bayesian analysis is MCMCMC!
- The heated chains use powers of the likelihood ratio in their acceptance ratio
- We must
**only**use the main chain to calculate posterior probabilities. - In reality, all these factors involve very large numbers. It's not uncommon to throw away thousands of trees as part of the burn-in and calculate the posterior from millions of trees.
- Heated chains: usually 4-5.
- MrBayes by Huelsenbeck is the main program in current use

## See also

## References

- Altekar G, Dwarkadas S, Huelsenbeck JP, and Ronquist F (2004). Parallel Metropolis coupled Markov chain Monte Carlo for Bayesian phylogenetic inference.
*Bioinformatics*. - Geyer CJ (1991). Markov chain Monte Carlo maximum likelihood. In Keramidas (ed.),
*Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface*. Fairfax Station: Interface Foundation, pp. 156-163.