By Adnan Darwiche
This publication presents a radical creation to the formal foundations and sensible functions of Bayesian networks. It presents an intensive dialogue of recommendations for development Bayesian networks that version real-world events, together with concepts for synthesizing types from layout, studying versions from info, and debugging versions utilizing sensitivity research. It additionally treats targeted and approximate inference algorithms at either theoretical and sensible degrees. the writer assumes little or no history at the lined topics, offering in-depth discussions for theoretically vulnerable readers and adequate useful information to supply an algorithmic cookbook for the approach developer.
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Extra info for Modeling and Reasoning with Bayesian Networks
Independence satisfies other interesting properties that we explore in later chapters. Independence provides a general condition under which the belief in a conjunction α ∧ β can be expressed in terms of the belief in α and that in β. Specifically, Pr finds α independent of β iff Pr(α ∧ β) = Pr(α)Pr(β). 13) is viewed as a consequence. 14) when we want to stress the symmetry between α and β in the definition of independence. It is important here to stress the difference between independence and logical disjointness (mutual exclusiveness), as it is common to mix up these two notions.
Given this method of specifying evidence, computing the new state of belief Pr can be done along the same principles we used for Bayes conditioning. In particular, suppose that we obtain some soft evidence on event β that leads us to change our belief in β to q. Since this evidence imposes the constraint Pr (β) = q, it will also impose the additional constraint Pr (¬β) = 1 − q. Therefore, we know that we must change the beliefs in worlds that satisfy β so these beliefs add up to q. We also know that we must change the beliefs in worlds that satisfy ¬β so they add up to 1 − q.
Edu/c2d/. 1. Show that the following sentences are consistent by identifying a world that satisfies each sentence: (a) (A =⇒ B) ∧ (A =⇒ ¬B). (b) (A ∨ B) =⇒ (¬A ∧ ¬B). 2. Which of the following sentences are valid? If a sentence is not valid, identify a world that does not satisfy the sentence. (a) (A ∧ (A =⇒ B)) =⇒ B . (b) (A ∧ B) ∨ (A ∧ ¬B). (c) (A =⇒ B) =⇒ (¬B =⇒ ¬A). 3. Which of the following pairs of sentences are equivalent? If a pair of sentences is not equivalent, identify a world at which they disagree (one of them holds but the other does not).