2 nd Cornell Probability Summer School 2006

A diffusion process model of the frequency of a mutation • Reversibility of a 1-dimensional diffusion process • Frequency spectrum and age of a mutation General binary coalescent trees • Combinatorial derivation of the age of a mutation • Ewens' sampling formula, a combinatorial derivation • Coalescent lineage distributions Gene trees and Coalescent trees • DNA sequences and the infinitely-many-sites model • Mutation histories, gene trees and coalescent trees • Ancestral inference from gene trees Importance sampling on coalescent histories • Constructing importance sampling algorithms • Examples of particular models and ancestral inference The Ancestral Recombination graph • Graphical description • Probability calculations on the graph • An MCMC algorithm for the time to the most recent ancestor along sequences A diffusion process model of the frequency of a mutation • Diffusion process model of the frequency of a mutation • Reversibility of a 1-dimensional diffusion process • Simulation of diffusion paths • Frequency spectrum and age of a mutation The population frequency of a mutation The frequency {X(t), t ≥ 0} is modelled by a diffusion process with generator L = 1 2 σ 2 (x) ∂ 2 ∂x 2 + µ(x) ∂ ∂x σ 2 (x) = x(1 − x) , µ(x) = β(x)x(1 − x) Denote ∆X(t) = X(t + ∆t) − X(t) E(∆X(t) | X(t) = x) = µ(x)∆t + o(∆t) Var(∆X(t) | X(t) = x) = σ 2 (x)∆t + o(∆t)