Evolution is change in heritable traits of biological populations over successive generations.
Evolutionary processes give rise to biodiversity at every level of biological organisation, including species, individual organisms, and genes.
A phylogenetic tree is a branching diagram or "tree" showing the evoluionary relationships between various species, individuals, or genes.
Populations evolve, individuals do not!
The random fluctuation in the frequency of a trait or allele (gene variant) distribution within a population due to the demographic process.
The probability per generation that a new allele fixes is therefore $\mu$: independent of population size!
This is not exactly true.
Given a set of $M$ sampled molecular sequences, infer the evolutionary relationships between them.
In a sense this is a just a clustering problem...
Single parameter is the speciation rate $\lambda$.
How do we go about combining all of these different pieces of information?
The answer is...
For propositions $A$ and $B$, the probability $P(A|B)$ is the degree to which $A$ is believed to be true,
on the condition that $B$ is true.
That's it! You can now do Bayesian statistics.
Mechanically applying the product rule yields
$$\color{darkred}{P(\theta_M|D,M)}=\frac{\color{yellow}{P(D|M,\theta_M)}\color{darkblue}{P(\theta_M|M)}}{P(D|M)}$$
with terms named the posterior, likelihood and prior.
The goal of phylogenetic inference is generally to learn about the phylogeny of a set of sampled genetic sequences.
As Bayesians we accomplish this by seeking to characterise the joint posterior: $$P(T,\mu,\theta|D) = \frac{1}{P(D)} P(D|T,\mu)P(T|\theta)P(\mu)P(\theta)$$
The tree likelihood $P(D|T,\mu)$ is the probability that the observed sequences were produced by evolving a single ancestral sequence down $T$ under a neutral substitution model with parameters $\mu$.
There is an efficient algorithm for computing this (summing over all possible ancestral sequences): Felsenstein's pruning algorithm.
The tree prior $P(T|\theta)$ is the probability for the tree $T$ based solely on the genealogical model and its parameters $\theta$.
(May also be considered the likelihood for the genealogical model parameters $\theta$ given the tree.)
MCMC (aka the Metropolis-Hastings algorithm) allows us to generate samples from a target distribution $P(\vec{x})$ assuming that we can evaluate $\alpha P(\vec{x})$ where $\alpha$ is an arbitrary unknown constant.
MCMC works by performing a biased random walk over $\vec{x}$ constructed so that the relative frequency of visiting state $\vec{x}$ at equilibrium is equal to $P(\vec{x})$.
Flexible, yet extremely computationally intensive!
BEAST is a free (as in freedom) software package that uses MCMC to perform Bayesian phylogenetic inference.
Project website: beast2.org
Tutorial 1:
Divergence time dating.
Tutorial 2:
Bayesian skyline plot.