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Unformatted text preview: Maximum likelihood II Peter Beerli October 3, 2005 1 Conditional likelihoods revisited Some of this is already covered in the algorithm in the last chapter, but more elaboration on the practical procedures and why are using these might give an even better understanding. This section follows the Felsenstein book pages 251255. We express the likelihood of the tree in Figure 1 as Prob( D ( i )  T ) = X z X y X w X x Prob( A,A,C,G,G,w,y,x,z  T )  T ) (1) where T = ( t 1 ,t 2 ,t 3 ,t 4 ,t 5 ,t 6 ,t 7 ,t 8 ). Each summation is over all 4 nucleotides. The above proba bility can be separated into Prob( A,A,C,G,G,w,y,x,z  T )  T ) =Prob( z ) Prob( w  y,t 3 )Prob( A  w,t 1 )Prob( A  w,t 2 ) Prob( y  t 5 ,z )Prob( C  y,t 4 ) Prob( x  z,t 6 )Prob( G  x,t 7 )Prob( G  x,t 8 ) Prob( z ) at the root is often assumed to depend on the stationary base frequencies. All parts are easy to calculate and if we order the terms of the sum in formula 1 and move the summations as far right as possible we get a summation pattern that uses the same structure as our tree ((C, (A, A)),(G, G)) Prob( D ( i )  T ) = X z Prob( z ) X y Prob( y  t 5 ,z )Prob( C  y,t 4 ) " X w Prob( w  y,t 3 )Prob( A  w,t 1 )Prob( A  w,t 2 ) #! X x Prob( x  z,t 6 )Prob( G  x,t 7 )Prob( G  x,t 8 ) ! 1 BSC5936Fall 2005 Computational Evolutionary Biology Figure 1: A tree with branch length and data This method was introduced by Felsenstein in 1973 as pruning but was known before that in pedigree analyses and even long before that as a summation reduction device called Horners rule. Even though it is named after William George Horner, who described the algorithm in 1819, it was already known to Isaac Newton in 1669 and even to the Chinese mathematician Chin Chiu Shao around 1200s (Horners rule Wikipedia.com 2005). Using the tree structure and calculating quantities (conditional likelihoods) on nodes we arrive at the solution outlined in the algorithm in the last lecture. 2 Scaling of likelihoods With large trees the conditional likelihoods will be very small and on finite machines we need to scale the conditional likelihoods so that they do not underflow. This would have dire consequences 2 BSC5936Fall 2005 Computational Evolutionary Biology because when all conditional likelihoods in a node approach zero, all following node in the downpass will be zero, independent of the values of their sister nodes. The general scheme to solve underflowwill be zero, independent of the values of their sister nodes....
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 Spring '08
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 Evolution

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