Using the EM algorithm to find the recombination values with the maximal likelihood for given evidence. Submitted by : Galia Shlezinger and Liat Ramati under the instruction of: Prof. Dan Geiger and Maayan Fishelzon

The Expectation Maximization algorithm Expectation: the expectation of the recombination values was computed using the BTE algorithm to compute the probability of the pedigrees selectors. Maximization : maximization was computed by dividing the expected number of recombinant meioses and dividing in by the total number of meioses.

The BTE Algorithm A tree of buckets, each bucket has only one parent. Each bucket in the tree represents a non- const and not eliminated variable in the baysian network of SuperLink.

The BTE algorithm - diagram First phase Second phase

BTE - struct bucket Id origvar vararr messages parent child Struct bucketThe original variables from the baysian net the variable represents The buckets messages : each message holds a function and the bucket it was sent from The variable in the bucket’s functions which are not in origvar

Optimizations Bucket merging : a bucket was merged with it’s father in the tree to avoid multiplying large probability functions, the bucket to merge are chosen according to “sepsize” which is a function of the number of variables in the seperator and the numValues of these variables. The bucket product hold a sorted list of buckets according to sepsize and each itaration of merge merges the first bucket from that list with it’s father.

Merging buckets

Optimization Cont. Setting variables as const when multiplying: in order to save space when multiplying functions with large probability functions some of the variables were set as const and for each const value possible for these variables the rest of the variables were multiplied, then all the results were added to get the result of the multiplication.