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Consider the following:

Consider the following:. P(b). P(h). H B E L F D. P(e|h) P(e|~h). P(l|h,b) P(l|~h,b) P(l|h,~b) P(l|~h,~b). P(f|e) P(f|~e). P(d|l) P(d|~l). P( d|h ) = P( d,h )/P(h)

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Consider the following:

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  1. Consider the following: P(b) P(h) H B E L F D P(e|h) P(e|~h) P(l|h,b) P(l|~h,b) P(l|h,~b) P(l|~h,~b) P(f|e) P(f|~e) P(d|l) P(d|~l) P(d|h) = P(d,h)/P(h) = [P(d,l,b,h) + P(d,l,~b,h) + P(d,~l,b,h) + P(d,~l,~b,h)] / P(h) = P(d|l)P(l|h,b)P(b) + P(d|l)P(l|h,~b)P(~b) + P(d|~l)P(~l|h,b)P(b) + P(d|~l)P(~l|h,~b)P(~b) = = P(b)[P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] + P(~b)[P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b)] P(d|l)[P(l|h,b)P(b) + P(l|h,~b)P(~b)] + P(d|~l)[P(~l|h,b)P(b) + P(~l|h,~b)P(~b)]

  2. Consider the following: f(B) P(b) , P(~b) f(H) H B E L F D P(l|h,b) P(l|~h,b)P(~l|h,b) P(~l|~h,b) P(l|h,~b) P(l|~h,~b)P(~l|h,~b) P(~l|~h,~b) f(L,h,B) f(E,H) f(F,E) f(d,L) = [P(d|l), P(d|~l)] P(d|h) = P(d,h)/P(h) = [P(d,l,b,h) + P(d,l,~b,h) + P(d,~l,b,h) + P(d,~l,~b,h)] / P(h) = P(d|l)P(l|h,b)P(b) + P(d|l)P(l|h,~b)P(~b) + P(d|~l)P(~l|h,b)P(b) + P(d|~l)P(~l|h,~b)P(~b) = = P(b)[P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] + P(~b)[P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b)] = ΣBP(B) ΣLP(d|L)P(L|h,B) = ΣB f(B) ΣL f(d,L)f(L,h,B) P(d|l)[P(l|h,b)P(b) + P(l|h,~b)P(~b)] + P(d|~l)[P(~l|h,b)P(b) + P(~l|h,~b)P(~b)] = ΣL P(d|L) ΣB P(L|h,B)P(B)

  3. Consider the following: f(B) P(b) , P(~b) f(H) H B E L F D P(l|h,b) P(l|~h,b)P(~l|h,b) P(~l|~h,b) P(l|h,~b) P(l|~h,~b)P(~l|h,~b) P(~l|~h,~b) f(L,h,B) f(E,H) f(F,E) f(d,L) = [P(d|l), P(d|~l)] P(b)[P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] + P(~b)[P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b)] = ΣB f(B) ΣL f(d,L)f(L,h,B) f(d,L,h,B) = f(d,L)f(L,h,B) f(d,h,B) ΣL f(d,L)f(L,h,B) f(L,h,B) f(B) f(B)f(d,h,B) f(d,L) P(l|h,b) P(l|h,~b) P(~l|h,b) P(~l|h,~b) P(d|l)P(l|h,b) P(d|l)P(l|h,~b) P(d|~l)P(~l|h,b) P(d|~l)P(~l|h,~b) [ P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] [ P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b) ] P(b)[P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] P(d|l) P(d|~l) P(b) P(~b) P(~b)[P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b)] then sum over different B Values  Sum over different L values for same B values

  4. Consider the following: f(B) P(b) , P(~b) f(H) H B E L F D P(l|h,b) P(l|~h,b)P(~l|h,b) P(~l|~h,b) P(l|h,~b) P(l|~h,~b)P(~l|h,~b) P(~l|~h,~b) f(L,h,B) f(E,H) f(F,E) f(d,L) = [P(d|l), P(d|~l)] … = P(d|l)P(l|h,b)P(b) + P(d|l)P(l|h,~b)P(~b) + P(d|~l)P(~l|h,b)P(b) + P(d|~l)P(~l|h,~b)P(~b) = = P(b)[P(d|l)P(l|h,b) + P(d|~l)P(~l|h,b)] + P(~b)[P(d|l)P(l|h,~b) + P(d|~l)P(~l|h,~b)] = ΣB f(B)ΣL f(d,L)f(L,h,B) P(d|l)[P(l|h,b)P(b) + P(l|h,~b)P(~b)] + P(d|~l)[P(~l|h,b)P(b) + P(~l|h,~b)P(~b)] = ΣL f(d|L)ΣB f(L,h,B)f(B) 1. Eliminate L (factor FL1 (B)) 1. Eliminate B (factor FB2 (L)) 2. Eliminate B (factor FB1 ()) 2. Eliminate L (factor FL2 ())

  5. Consider the following: f(B) P(b) , P(~b) f(H) H B E L F D P(l|h,b) P(l|~h,b)P(~l|h,b) P(~l|~h,b) P(l|h,~b) P(l|~h,~b)P(~l|h,~b) P(~l|~h,~b) f(L,h,B) f(E,H) f(F,E) f(d,L) = [P(d|l), P(d|~l)] P(d) = [P(d,l,b,h) + P(d,l,~b,h) + P(d,~l,b,h) + P(d,~l,~b,h) + P(d,l,b,~h) + P(d,l,~b,~h) + P(d,~l,b,~h) + P(d,~l,~b,~h) = P(d|l)P(l|h,b)P(b)P(h) + P(d|l)P(l|h,~b)P(~b)P(h) + P(d|~l)P(~l|h,b)P(b)P(h) + P(d|~l)P(~l|h,~b)P(~b)P(h) + P(d|l)P(l|h,b)P(b)P(~h) + P(d|l)P(l|h,~b)P(~b)P(~h) + P(d|~l)P(~l|h,b)P(b)P(~h) + P(d|~l)P(~l|h,~b)P(~b)P(~h) = = = ΣHP(H) * ΣBP(B) * ΣLP(d|L)P(L|H,B) FH1() FB1(H) FL1(H,B) ΣHP(H) * ΣLP(d|L) * ΣBP(L|H,B)P(B) FH2() FL2(H) FB2(H,L) ΣBP(B) * ΣLP(d|L) * ΣHP(L|H,B)P(H) FB3() FL3(B) FH3(B,L)

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