BNFO 602 Phylogenetics

1. BNFO 602 Phylogenetics Usman Roshan

2. Maximum Likelihood • D = data, M = model • Bayes rule P(M|D) = P(D|M)P(M) / P(D) P(M|D) is the posterior probability P(D|M) is the likelihood P(M) is the prior probability on the model By rewriting P(D) we get = P(D|M)P(M) / ∑M P(D|M)P(M) which implies that P(M|D) is proportional to P(D|M)P(M) Note that by assuming uniform priors P(M|D) = P(D|M)1/k / ∑M P(D|M)1/k

3. Maximum Likelihood • Data (input) is the alignment • Model consists of • the tree with branch lengths and leaves labeled with the DNA sequences in the data (input) • a DNA sequence evolution model (such as Jukes Cantor) • How do we compute the likelihood P(D|T) of the tree below?

4. Which of the two trees below have the higher likelihood?

5. Maximum Likelihood • ML problem: Under a fixed model find the tree with branch lengths and internal nodes that has the highest likelihood. • Very large search space • NP-hard • Sub-problems • What is the likelihood of a tree with branch lengths and internal nodes? • Linear time solution • What if no internal nodes are given? • Felsenstein’s algorithm gives linear time solution • What if no branch lengths are given? • NP-hard • We use gradient descent

6. Maximum Likelihood • Comparison to MP: • Both are NP-hard • For fixed tree it takes polynomial time to find the parsimony score • For fixed tree is is NP-hard to find the likelihood score • Similar local search heuristics as MP