Metropolis-coupled Markov chain Monte Carlo
From Christoph's Personal Wiki
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". Only the information from the cold chain is recorded. Periodically, trees between chains may be swapped.
(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
