We begin with a randomly selected tree, which is well defined, which has a likelihood and prior. We then make a small change to the tree. If the new proposed tree has a greater prior x likelihood function
The Markov chain usually begins with a randomly chosen value, and so in analysis the first ~10% of generations are discarded as "burn-in".
Criticism of Bayesian Phylogenetics
Bayesian phenogenetics is by no means perfect though, despite uncertainty is present in your result, you have to be state your prior probability; and in situations where that is unknowable (for example language phylogenies) you have to invent a prior probability, which can skew your result if it's poorly picked. While these unknowns can be aided by using a prior distribution as opposed to a prior probability,
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 islands was sporadic, and in the case of Easter Island, non existent (SOURCE). As noted before there is a striking similarity of 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, and migrating via the Philippines and New Guinea and reaching Melanesia by roughly 1400BCE and reaching Samoa by 900BCE. Entangled Bank theory however puts forward that there was no single 'express train' to Polynesia, it emphasises smaller migration as well as the long cultural and genetic interactions between the Polynesians, Melanesians and East Asians. 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, then suggests that upon reaching Melanesia migration slowed, and there was a long delay (leading to cultural and genetic admixture) before migration to Polynesia.
Contemporary linguists have attempted to solve these competing theories using Bayesian inference, and application of MCMC algorithms. An important paper being Language Phylogenies Reveal Expansion Pulses and Pauses in Pacific Settlement. The research builds on lessons learnt in the comparative method, using a relatively short word list (210), establishing cognates between 400 languages and creating a consensus tree using Bayesian inference, modelling language evolution based on single cognate gain/loss . The posterior probability distribution comprised of 4,200 trees, and the consensus tree provided more evidence of Dempwolff's original theory of the singular oceanic language group. The consensus tree also agreed with Grace et al's positioning of the Polynesian languages nested within the Central-Pacific subgroup. The research provides evidence for a Taiwanese origin of migration through Melanesia, and pause/pulse theory of migration. With an initial refractory period before rapid expansion out of Taiwan through Melanesia, with a second pause occurring once the Polynesians arrived at Samoa.
both this and Language trees support the express-train sequence of Austronesian expansion.
Summary diagram
From all of these pieces of data, I've attempted to create a cumulative map of Polynesian migration:
\cite{churchill1912} \cite{Walker_2010} \cite{Diamond_2003}\cite{Greenhill_2010}\cite{churchill1911}\cite{Gray_2009}\cite{2013}\cite{Gray_2000}