Maximum Likelihood

The likelihood function is the simultaneous density of the observation, as a function of the model parameters. L(  ) = Pr(Data|  ) If the observations are independent, we can decompose the term into

An example Consider the estimation of heads probability of a coin tossed n times Heads probability p Data = HHTTHTHHTTT L(p) = Pr(D|p) = pp(1-p)(1-p)p(1-p)pp(1- p)(1-p)(1-p) = p 5 (1-p) 6

L(p) = p 5 (1-p) 6 = 5/11

Maximum Likelihood Take the derivative of L with respect to p: Equate it to zero and solve: p = 5/11 ^

Log Likelihood For computational reasons, we maximise the logarithm lnL = 5 lnp + 6 ln(1-p) with derivative p = 5/11 ^

A tree

Tree likelihood: Assumptions 1.Evolution in different sites is independent. 2.Evolution in different lineages is independent.

Pr(A,C,C,C,G,x,y,z,w|T) = Pr(x) Pr(y|x,t 6 ) Pr(A|y,t 1 ) Pr(C|y,t 2 ) Pr(z|x,t 8 ) Pr(C|z,t 3 ) Pr(w|z,t 7 ) Pr(C|w,t 4 ) Pr(G|w,t 5 )

Using models Observed differences Actual changes AG CT Example: Jukes-Cantor, if i=j, if i≠j p t : proportion of different nucleotides

30 nucleotides from  -globin genes of two primates on a one-edge tree * * Gorilla GAAGTCCTTGAGAAATAAACTGCACACTGG Orangutan GGACTCCTTGAGAAATAAACTGCACACTGG There are two differences and 28 similarities tt lnL  t= lnL=