Joint conditional probability in bayes net

[lyuriedin]

Homework Statement

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)

Homework 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

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.

 

[LightHero]

Do I just take the terms P(a) and P(d|b) and multiply them together? If so, how do I go about summing over the other terms? Thanks for any help.