pi_lec8 - 20.181 Lecture 8 Contents 1 Probability of a tree...

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20.181 Lecture 8 Contents 1 Probability of a tree o 1.1 Marginalizing 2 Designing an Algorithm o 2.1 Remember Fibonacci! o 2.2 Efficient computation 3 Greedy Algorithm for trying trees o 3.1 Branch Lengths Probability of a tree How do we get from what we know (score of data given a tree) to what we want (score of the tree, given the data)? From last time, one of the assumptions that we made is that without knowing anything about the data, two trees (T1 and T2) are equally likely. Note: when we propose a tree, we'll also find the branch lengths that work best for that tree. We'll treat that as part of the topology. Expression for the joint probability of all data: P ( D | T ) = P ( A , C , C , C , G , x , y , z , w | T ) T = topology of the tree (includes t1,t2. .. all the branch lengths) We're going to refer to x,y,z,w as "nuisance parameters" We don't care what their values ARE, we're just going to try ALL of them, and just integrate through. Join probability of the data can be separated into each of the independent events (mutations), and the probability of each event is exactly what the J-C formula gives us the way to calculate!
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P ( A , C , C , C , G , x , y , z , w | T ) = P * P * P * P x y z w P ( x ) = P x P ( y | x , t ) * P ( A | y , t ) * P ( C | y , t ) = P 6 1 2 y P ( z | x , t ) * P ( C | z , t ) = P 8 3 z P ( w | z , t ) * P ( C | w , t ) * P ( G | w , t ) = P 7 4 5 w Now that we have an expression for the joint probability, we can sum over the nuisance paramaters. For each possible set of values for the nuisance parameters, the joint probability distribution will take on a particular value. We then ADD these because we don't care what the nuisance parameters' values are, they can be one set of values, OR another set of values, OR another set of values . .. (each sum
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pi_lec8 - 20.181 Lecture 8 Contents 1 Probability of a tree...

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