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Likelihood Computation  Given a Bayesian network and evidence e, compute P( e ) Sum over all possible values of unobserved variables Bet1Die Win1 e = { Win1 = true }

The Basic Concept P(e,Die=1)= P(e,Die=3)= P(e,Die=5) and P(e,Die=2)= P(e,Die=4)= P(e,Die=6) Bet1Die Win1 Val(Bet1) = {odd,even} e = { Win1=true }  The exact value of Die need not be known to calculate exact likelihood  Group values, calculate once for each group

Value Abstraction Val(Die)  Val(Die a ) {1,4} {2,5,6} {3} A partition of a variable’s domain Bet1Die Win1

Safe Value Abstraction An abstraction is safe w.r.t. evidence e if Preserves likelihood information Val(Die)  Val(Die a ) {1,3,5} {2,4,6} Bet1Die Win1

Win 2 Bet2 Val(Bet2)={ 1-2, 3-6 } {1,3,5} {2,4,6} {1} {2} {3,5} {4,6} {1,3,5} {2,4,6} e = {Win1=true}e = {Win1=true, Win2=true} Bet1Die Win1 Safe Value Abstraction Win1 Val(Bet1)={odd, even } Win 2 A safe abstraction for Val(Die)  Need to refine Refinement

Cautious Value Abstraction Bet1Die Win1 Win 2 Bet2 Maximal abstraction - a tight refinement {2} {3,5} {4,6} {1} {1,3,5} {2,4,6} {1,2} {3-6} Win1=trueWin2=true Val(Bet1)={ odd, even } Val(Bet2)={ 1-2, 3-6 }

Abstracting a Bayesian Net  An abstraction of X i implies a partition of Pa i ’ s values  Abstract each variable after it’s children are abstracted, use a tight refinement of all partitions implied by children Output - G a : For each variable: 1. Calculate maximal abstraction 2. Propagate to parents X Initialization: Abstract observed variables  Linear in # variables and network representation

The Application-Genetic Linkage Analysis u The goal - Find the location of a disease (target) gene on a chromosome relative to some other (known) locations Map of human chromosome 16 Known loci

Recombination Fraction AB No crossover between A and BCrossover between A and B P =  P = 1- 

u The data - pedigrees Linkage Analysis /3

u Converting the pedigree to a Bayesian net One locus: The Probabilistic Model Orange - genotype nodes Blue - phenotype nodes Red - selector nodes (these represent linkage)

Locus # Locus #2 The Probabilistic Model u More than 1 locus: s2s2 s1s1

1e+6 1e e e+51e+7 Abstracted Original Clique-tree size Abstracted Original Network size Experimental Evaluation u 90 pedigrees (5-200 individuals) from 10 studies u Total of 280 linkage analysis problems Varied number of loci # loci:

Bet1Die Win 2 Bet2 Win1 Abstracting Multiple Variables {1,3,5} {2,4,6} Die a odd even odd even Bet1 a loss win 1,o 2,o 3,o 4,o 5,o 6,o 1,e 2,e 3,e 4,e 5,e 6,e

Clique-Tree Elimination X,V,U X,W X,V,Z V,Y C1C1 C2C2 C3C3 C4C4

Message-Specific Abstraction X,V,U X,W X,V,Z V,Y C1C1 C2C2 C3C3 C4C4 Given safe abstractions for f 3, m 1  3, m 2  3 - construct a safe abstraction for m 3  4 Refinement  multiplication Projection  summation  is safe for message m if Use dynamic programming to efficiently compute a safe abstraction for the whole tree

Experimental Evaluation u How much more can we save ? # loci: Abstracted clique-tree Abstracted network e e+6 Clique-tree size ratio Abstracted network e+6 Clique-tree sizeRatio

Total Reduction Clique-size Ratio (orig/abs) Problem size (#individuals X #genotypes) e+006 1e+007 1e+008 1e

Summary u Safe abstraction w.r.t. specific evidence u An algorithm to reduce problem complexity H Linear in net representation H Independent of inference procedure Motivated by VITESSE[ ] u Further reductions with inference procedure known u Caveats l As costly as inference H Cost is ammortized when used for e.g. parameter estimation l Representation of abstractions