20.181/Lecture7

quick comment on upPass



 * not necessary to find the best tree (you won't be tested on it)
 * but here's the correct way to do it:
 * (from Peter Beerli's website)

definitions

 * Fx: the upPass set we want to get to
 * Sx: the downpass set we got to
 * 1) ancestor = a
 * 2) parent = p, node we're looking at
 * 3) children = q,r



Revisit overall strategy
 for all possible trees:
 * Although up until now we've always started with a tree of known topology, a lot of times you wouldn't know the tree topology beforehand
 * compute score (tree)

return best tree 

Scoring functions

 * 1) max parsimony (fewest mutations)
 * 2) generalized parsimony (Sankoff: weighted mutation costs)
 * 3) Maximum Likelihood

ML intro

 * examples of a ML estimator:
 * for normally distributed random var X, X(bar), the mean of the data you observe, is a ML estimator of the mean of the distribution they were drawn from
 * A best fit line thru data is a ML estimator.



Probability Refresher
total area of a box = 1

 p(A)= 0.3, p(B)= 0.3 p(A,B)= 0.1 p(A|B)= 0.1 / (0.1+0.2) = 1/3 = p(A,B) / p(B) p(B|A) = 0.1 / (0.1+0.2) = p(A,B) / p(A) With a little manipulation we can derive Bayes' Rule: p(A|B) = p(B|A) * p(A) / p(B)



ML in trees

 * We are looking for the best tree, given some data. What is the best tree T given the data D?
 * p(T|D) is what we want to maximize
 * Not obvious how we want to do that... use Bayes Law to rearrange into something we can intuitively understand

 p(T|D) = p(D|T) * p(T) / p(D) 


 * p(D) is a constant ... we don't have to worry about it
 * What is p(T), the a priori probability of the tree ?
 * Well, without looking at the data, do we have a way of saying any tree is more likely than another one if they don't have any data associated with them ?
 * No... not really


 * So what we're left maximizing is just p(D|T) and that sounds like a familiar concept!

NOTE: Tree now consists of topology AND distances We ask, what is the probability of each mutation occuring along a branch of a certain length? What is the probability that they ALL occurred, to give us the sequences we see today?

 p(D|T) = p(x->A|d_1) * p(x->y|d_2) * p(y->G|d_3) * p(y->G|d_4) p(A U B) = p(A) + p(B) - p(A,B) p(A B) = p(A)*p(B) 


 * We treat all of these mutations along the different branches as independent events (that's why you multiply the probabilities, because all the events have to happen independently.)

Jukes-Cantor

 * based on a simple cost "matrix"
 * probability of changing from one particular nucleotide to another particular nucleotide is 'a'
 * probability of any nucleotide staying the same is '1-3a'

 if x == y :
 * [JC eqn you'll derive in the hw]

if x != y :
 * [JC eqn you'll derive in the hw]



Evolutionary Model
gives us likelihood of (D|T) (need branch lengths)

 downPass for ML
 * compute L(p|q,r,d)



q, r = likelihood of the two subtrees, d are the distances to them