Equation (14.1) on page 513 defines the joint distribution represented by a Bayesian network in terms of
Question:
a. Consider a simple network X Y Z with three Boolean variables. Use Equations (13.3) and (13.6) (pages 485 and 492) to express the conditional probability P(z | y) as the ratio of two sums, each over entries in the joint distribution P(X, Y, Z).
b. Now use Equation (14.1) to write this expression in terms of the network parameters θ(X), θ(Y |X), and θ(Z | Y ).
c. Next, expand out the summations in your expression from part (b), writing out explicitly the terms for the true and false values of each summed variable. Assuming that all network parameters satisfy the constraint xi θ(xi | parents(Xi)) = 1, show that the resulting expression reduces to θ(x | y).
d. Generalize this derivation to show that θ(Xi |Parents(Xi)) = P(Xi |Parents(Xi)) for any Bayesian network.
Equation (14.1)
Equations (13.3)
Equations (13.6)
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Artificial Intelligence A Modern Approach
ISBN: 978-0136042594
3rd edition
Authors: Stuart Russell, Peter Norvig