Bayesian Network Problem: What is P(A=T, B=T, C=F, D=F)? Suppose you have these probabilities in a bayesian network. What is P(A=T, B=T, C=F, D=F)?
I attempted to answer this by saying that P(A,B,C,D) = P(A) P(B|A) P(C|A,B) P(D|A,B,C), according to the Chain Rule. But it doesn't really seem to work, because D needs to be conditional on E for it to work. So I really need help figuring this out. Thanks!

 A: Factorised distribution of the network
P(E)* P(A)* P(B|A)* P(C|B)* P(D|B, E)

Look at table P(D|B, E) - remove rows inconsistent with evidence, leaving
D B E P
F T T 0.8
F T F 0.7

You can sum out E my multiplying this by the prior of E
So altogether: read off the other values from the cpt's and multiply out E
# P(A=T)* P(B=T|A=T)* P(C=F|B=T)* (P(D=F|B=T, E)*P(E))
     0.6*    0.2*        0.3*     (0.8*0.9 + 0.7*0.1)
#0.02844


You can check this using the gRain package in R which uses the clique tree propagation algorithm (ref)
library(gRain)

# define cpt's
tf <- c("T","F")
a <- cptable(~a, values=c(6, 4),levels=tf)
e <- cptable(~e, values=c(9, 1),levels=tf)
b.a <- cptable(~b|a, values=c(2, 8, 4, 6),levels=tf)
c.b <- cptable(~c|b, values=c(7, 3, 95, 5),levels=tf)
d.be <- cptable(~d|e:b, values=c(2, 8, 3, 7, 1, 9, 6, 4),levels=tf)

plist <- compileCPT(list(a, e, b.a, c.b, d.be))

# check tables are input correctly
plist$b
plist$d

# build network
net <- grain(plist)
plot(net)


# set evidence
ev <- setEvidence(net, nslist=list(a="T", b="T", c="F", d="F"))

# probability of this evidence
pEvidence(ev)
#[1] 0.02844

(ref) Local Computations with Probabilities on Graphical Structures and Their Application to Expert Systems, S. L. Lauritzen and D. J. Spiegelhalter
A: I am very new to graphical models and would love to know how to solve such a problem. Here is my line of thoughts.
The full joint probability is:
$P(A,B,C,D,E)=P(A)P(E)P(B|A)P(C|B)P(D|B,E)$
What is asked for is $P(A,B,C,D)$. To get it, you need to marginalize out E from the full joint probability. Something like this:
$P(A,B,C,D)=P(A)P(B|A)P(C|B)\sum _E P(E)P(D|B,E)$
I don't know if my line of reasoning is correct, and if it is correct, I am not sure what to do with that sum and how to simplify it.
