1. 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

2. 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, labeled 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.

3. 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 ).

4. 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.

5. 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 that can be optimized.

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

7. C. Minimum Evolution

8. 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).

9. 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.