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7]. 3, 1]. To determine the value of c, a random number α is generated from a uniform distribution over [0, 1]. 4. Because it falls into the bin bin(c = 1), the value of c is set to 1. Similarly, we can obtain the values for other root variables, say (a = 0, b = 1, c = 1, g1 = norm, g2 = norm, g3 = norm, g4 = ab). To determine the values for remaining variables, we start with a variable such that the values of its parents have been determined, say, d. 2 is generated. The value of d is then set to 0.
Many useful properties of probability distributions can be proven from the preceding axioms. One such property is Bayes’s rule: P(Y |X, Z )P(X |Z ) . 6) where x ∈ D X and y ∈ DY . We often need to derive the distribution over X from the distribution over Y ⊃ X . 7) w∈DW where X ⊂ Y , W = Y \X , x ∈ D X , and z ∈ D Z . The operator\denotes set difference. For simplicity, we write P(X, W |Z ) = P(X |Z ) = W P(Y |Z ), W and the subset W of variables is said to be marginalized out. The purpose of representing knowledge over a problem domain V using a probability distribution is to be able to reason about the state of the domain given some observations.
As we shall see, they allow effective acquisition and inference in many practical domains. The word effective is used loosely in this book to refer to a method or algorithm that is efficient on an average input instance but can be intractable in the worst case. 4 Graphs The fundamental idea underlying effective representation and inference with probabilistic knowledge is that in the real world not every variable is directly dependent on every other variable. The output of a digital gate is directly dependent on its input and the state of the gate.