The most commonly used version of an MCMC method in phylogeny is the Metropolis-Hastings algorithm. The mathematics of the method is complicated but it involves taking many samples from the posterior probability distribution over a period of generations until you can approximate the posterior probability distribution, without having to calculate the distribution directly. The algorithm begin with a randomly selected tree, which is well defined. If little is initially known about the phylogeny before computation a sensible prior would be uniform probability distribution for all available trees. This tree is compared to one of the other possible trees by the ratio of the new tree over the old (This is how the marginal likelihood is excluded, as it cancels during division). If the new proposed tree has a greater prior x likelihood function it is selected, and a new tree is chosen. As the Markov chain usually begins with a randomly chosen value, the first ~10% of generations are discarded as "burn-in" until the algorithm "forgets" its initial tree. Over thousands of generations, and hundreds of samples an accurate tree can be selected with some certainty.
Parsimony, Bayesian phylogenetics & Polynesian migration
Bayesian phylogenetics are particularly useful in the field of Polynesian language evolution due to the unique migratory lifestyle of Polynesian peoples. This largely eliminates some of comparative method criticism of tree modelling as contact between distal Polynesian islands was sporadic, and in the case of Easter Island, non existent\cite{Drechsel_2014}. As noted before there is a striking similarity of the vocabulary in the Polynesian languages, which also assists in the effectiveness of biological models. As migration through Polynesia has largely been established, and X biological applications of linguistic phylogeny in Polynesia apply to finding the homeland of the Polynesian peoples, and the speed of their migration through Melanesia.
Current theories of Polynesian people's origin fall into two main camps: Express Train theory and the Entangled Bank theory \cite{Kayser_2000}. The more popular theory, Express Train, first put forward by Jared Diamond \cite{Diamond1988} posits that Polynesian migration originated in Taiwan and started relatively recently 3000-1000BCE. Migration took place via the Philippines and New Guinea, reaching Melanesia by roughly 1400BCE and reaching Samoa (Western Polynesia) by 900BCE. Entangled Bank theory, on the other hand, puts forward that there was no single 'express train' to Polynesia, it emphasises many smaller migration events, as well as the long cultural and genetic interactions between the Polynesians, Melanesians and East Asians. Some newer theories such as Kayser et al.'s Slow Boat theory attempts to marry these two ideas, while supporting a Tawainese origin and rapid migration to Melanesia, Slow Boat theory suggests that upon reaching Melanesia migration slowed, and there was a long delay (leading to cultural and genetic admixture) before migration to Polynesia.
Biological models were first applied to the Polynesian migration question in the 2001 paper Language trees support the express-train sequence of Austronesian expansion. The author, Russell Gray, used a simple system of character mapping based on maximum parsimony to establish the sequence of colonisation events that led to the colonisation of Polynesia. Languages were given a certain ranking based on their geographic position, and that geographic positions relation to the suspected order of migratory events \cite{Gray_2001}.