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Joint conditional probability in bayes net

  1. Oct 4, 2012 #1
    1. The problem statement, all variables and given/known data

    We are given a belief network such that the overall joint probability after accounting for conditional independence is:

    P(a,b,c,d,e) = P(a)P(c|a,b)P(d|b)P(e|d,b)
    The values provided are:

    P(a = false) = .02
    P(b = false) = .95
    P(c= false|a=true, b=true) = .97
    P(c= false|a=true, b=false) =.04
    P(c= false|a=false,b= true) = .99
    P(c= false|a=false,b=true) = .1
    P(d= false|b=true) = .03
    P(d= false|b=true) = .98
    P(e=false|d=true, b= true) = .92
    P(e=false|d=true, b= false) = .01
    P(e=false|d=false, b= true) = 1
    P(e= false|d= false, b= false) = 1

    The problem is to find

    P (b=true|e=false)

    2. Relevant equations

    I know that

    P(b|d) = P(b,d)/P(d)

    and so I need to sum out until I get the marginals I need

    3. The attempt at a solution

    i.e. P(b,d)=Sum_over_a(Sum_over_c(Sum_over_e(P(a,b,c,d,e)))

    = Sum_over_a(Sum_over_c(Sum_over_e(P(a)P(c|a,b)P(d|b)P(e|d,b)))

    But I can't work out how to do this.
  2. jcsd
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