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Some sample
problems
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(a) What is the probability that all five of these Boolean variables are simultaneously true?
(b) What is the probability that all five of these Boolean variables are simultaneously false?
(c) What is the probability that A is false given that the four other variables are all known to be
true?
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(a) Write down all the independencies not conditioned on other
variables that are enforced by this Bayesian network, using the
notation A B to mean that A is independent of B.
(b) Write down three independencies which do not necessarily hold in
this Bayesian network.
(c) Write down all the conditional independencies that are enforced by
this Bayesian network, using the notation A B|C to mean that A is
conditionally independent of B given C.
(d) Write down three conditional independencies which do not
necessarily hold in this Bayesian network.
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(a) What are the conditional probability distributions (CPTs) that are
represented in this Bayesian network?
(b) Write down the joint probability distribution over X, Y , and Z as
represented by this Bayesian network. This expression should be written
in terms of the CPTs you enumerated in (a).
(c) Write down an expression in terms of these CPTs for P(X,Z) (hint: sum
the variable Y “out” from the joint distribution you wrote above).
(d) Based on the expression in (c), and the definition of independence,
are X and Z independent?
(e) Write down an expression for P(X,Z|Y).
(f) Based on the expression in (e), and the definition of conditional
independence, are X and Z conditionally independent given Y ?
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(a) What are the conditional probability distributions (CPTs) that are
represented in this Bayesian network?
(b) Write down the joint probability distribution over X, Y, and Z as
represented by this Bayesian network. This expression should be written
in terms of the CPTs you enumerated in (a).
(c) Write down an expression in terms of these CPTs for P(X,Z) (hint: sum
the variable Y “out” from the joint distribution you wrote above).
(d) Based on the expression in (c), and the definition of independence,
are X and Z independent?
(e) Write down an expression for P(X,Z|Y).
(f) Based on the expression in (e), and the definition of conditional
independence, are X and Z conditionally independent given Y?
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(a)What is the probability that all four of these Boolean variables are
simultaneously true?
(b)What is the probability that C and B are true and D is false?
(c) What is the probability that C is true given that D is false and B is
true?
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Consider the Bayesian Network above. Assume that each of the
variables are boolean valued. For each of the following questions, state
the total number of operations (multiplications and additions) that it will
take to compute the answer. For example, if A and B are binary variables,
a P(B)P(a|B) will take two multiplications(P(B)P(A|B) and P(B)P(¬A|B))
and one addition (adding those two terms). Delete any summations that
are irrelevant to the query before computing your solutions.
Compute P(X3|X4) i.e., the probability that X3 is true given that X4 is true.
Assume that the variables are summed out in the order: X5, X1, X2.