Given the following Bayesian Network consisting of four Bernoulli random variables and the associated conditional probab...

Question

Given the following Bayesian Network consisting of four Bernoulli random variables and the associated conditional probability tables:

P(V=0) = 0.5, P(V=1) = 0.5

P(U=0|V=0) = 0.5, P(U=1|V=0) = 0.5
P(U=0|V=1) = 0.5, P(U=1|V=1) = 0.5

P(W=0|U=0) = 1,   P(W=1|U=0) = 0
P(W=0|U=1) = 0,   P(W=1|U=1) = 1

The value of P(U=1, V=1, W=1, Z=1) is ______ (rounded to three decimal places).

Accepted Answer

P(U=1, V=1, W=1, Z=1) = 0.125 We have a Bayesian network with four Bernoulli variables and given conditional probability tables (CPTs). To compute a specific joint probability, we use the chain rule factorized according to the network structure: multiply the CPT entries for each variable conditioned on its parents. The network structure (from the CPTs) shows V is a root, U depends on V, W depends on U, and Z depends on both V and W.