Lecture 6B – Optimality Criteria: ML & ME

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Presentation transcript:

Lecture 6B – Optimality Criteria: ML & ME LH = Pr (Data | Hypothesis) = P (D | H, M) L(t) = P (D | t, m) Just as in parsimony, we assume independence of characters. Single-site Likelihood

given the tree and model of evolution. Single-site Likelihood – Probability of a single column in the alignment, given the tree and model of evolution. Note, we’re indexing branches, labelled vx,y,, and we’re interested in their lengths (Units are substitutions per site, a function of rate x time). To calculate the SSL for this site, we sum the probabilities for all possible character-state reconstructions at each internal node. There are n-1 internal nodes, each of which could have one of 4 possible states. There are 4n-1 unique reconstructions: 4n-1 = 43 = 64.

Calculating Single-site Likelihoods (Sum probabilities of all reconstructions) r1 r2 r3 r64 or So now, the issue becomes how we calculate the Pr( Rr|t ).

Calculating Reconstruction Probabilities In reconstruction r, m is the state at node 3, k is the state at node 1, l is the state at node 2 Pr (Rr | t ) = pm x Pm,k(v3,1) x Pk,G(v1,w) x Pk,A(v1,x) x Pm,l(v3,2) x Pl,C(v2,y) x Pl,C(v2,z) pm is the frequency of the nucleotide m. This provides an estimate of the probability of observing state m at the root node. Pi,j is the probability of substitution between states i & j. This is derived the model of sequence evolution that we assume (much more on those in a few weeks). This is in some sense analogous to the step matrix that determines costs for transformations between states in the Sankoff algorithm in parsimony.

Calculating Reconstruction Probabilities Start at the root & traverse the tree Pr (Rr | t ) = pm x Pm,k(v3,1) x Pk,G(v1,w) x Pk,A(v1,x) x Pm,l(v3,2) x Pl,C(v2,y) x Pl,C(v2,z) So the SSL would be the sum of all possible reconstructions. Each summation is across all four nucleotides. Branch lengths (vi,j) are parameters to be optimized.

A few points to note: Site patterns The frequency of site pattern a. 1 A G T A C A . . . . . . . . . . . . . . . 2 A G T A . . . . . . . . . . . . . . . . . 3 A G T A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . n A G T A . . . . . . . . . . . . . . . . . Pattern 1 2 3 1 . . . . . . . . . . . . . . . .(a) Site patterns The frequency of site pattern a.

C. Minimum Evolution

Additivity Additivity: dAB = pAB = v1 + v2 , dAC = pAC = v1 + v3 + v4 , dAD = pAD = v1 + v3 + vd , dBC = pBC = v2 + v3 + v4 , dBD = pBD = v2 + v3 + v5 , dCD = pCD = v4 + v5 a usually equals 2 (so this is a least-squares method). wij allows weighting of the piar-wise error terms. wij = 1 assumes that the errors are identical across all dij. wij = 1/dij assumes that errors are proportional to dij. wij = 1/d2ij assumes that errors are proportional to the square root of dij. wij = 1/s2ij weights by the expected variance in the dij (which may not be known).

Relationships among Optimality Criteria Both MP and ML are character based (whereas ME is not). Both ME and MP minimize the amount of evolution (i.e., sum of branch lengths). Both ME and ML rely on an explicit model of sequence evolution.