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BAYESIAN NETWORKS CHAPTER#4 Book: Modeling and Reasoning with Bayesian Networks Author : Adnan Darwiche Publisher: CambridgeUniversity Press 2009

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Introduction Joint Probability Distribution can be used to model uncertain beliefs and change them in the face of Hard and Soft Evidence. Problem with JPD is that size grows exponentially with the number of variables which introduces modeling and computational difficulties.

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Need for BN BN is a graphical modeling tool for compactly specifying JPD BN relies on the basic insight that: “ independence forms a significant aspect of belief” “Elicitation is relatively easily using the language of graph”

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Example Earthquake (E) Burglary (B) Alarm (A) Radio (R) Call (C) BN is a Directed Acyclic Graph Nodes are Proposition al Variables Edges are Direct Causal Influences

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Example We would expect our belief in C to be influenced by some Evidence on R For example if we get a Radio report that an Earthquake took place then our belief in Alarm triggering would increase which would increase our belief in receiving call from a neighbor However we would not change our belief if we knew for sure that the Alarm did not trigger Thus C would be independent of R given ¬A

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Formal Representation of Independence Given a variable V in a DAG G: Parents (V) are the parents of V [Direct Causes of V] Descendants(V) are the set of variables N with a directed path from V to N [Effects of V] Non_Descendants(V) are the variables other that Parents and Descendants

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Independence Statement / Markovian Assumption

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Examples of Independence Statements I (C,A, {B,E,R} ) I (R,E, {A,B,C} ) I (A,{B,E}, R) I (B, ø, {E,R}) I (E, ø, B) Earthquake (E) Burglary (B) Alarm (A) Radio (R) Call (C)

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Parameterizing the Independence Structure Parameterizing means quantifying the dependencies between Nodes and their Parents In other words construction of CPT For every variable X in the DAG G and its parents U, we need to provide the probability Pr(x|u) for every value x of variable X and every instantiation u of parents U

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Formal Definition of Bayesian Network A Bayesian Network for variables Z is a pair where: G is a directed acyclic graph over variables Z called the Network Structure is a set of CPT’s one for each variable in Z called the Network Parameterization (X|U) would be used to denote the CPT for variable X and its parents U, and refer to the set XU as a Network Family.

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Def (continue..) denotes the value assigned by CPT to the conditional probability Pr (x|u) and call it Network Parameter Instantiation of all the network variables are called Network Instantiations Network parameterNetwork instantiation a a a (b|a) b ( ¬ c|a) ¬c (d|b, ¬ c) d ( ¬ e| ¬ c) ¬e

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Chain Rule for Bayesian Networks Network Instantiations z is simply the product of all network parameters compatible with z

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Properties of Probabilistic Independence Recall : I (X,Z,Y) Pr(x|z,y) = Pr(x|z) or Pr(y|z) =0 for all instantiations x,y,z Graphoid Axioms: Symmetry Weak Union Decomposition Contraction

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Symmetry If learning Y does not influence our belief in x then learning x does not influence our belief in y By Markov(G) we know that: I (A,{B,E},R) Using Symmetry: I (R,{B,E},A) Earthquake (E) Burglary (B) Alarm (A) Radio (R) Call (C)

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Decomposition If learning yw does not influence our belief in x then learning y alone or learning w alone does not influence our belief in x

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Weak Union If the information yw is not relevant to our belief in x then the partial information will not make the rest of the information relevant

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Contraction If learning the irrelevant information y the information w is found to be irrelevant to our belief in x then the combined information must have been irrelevant from the beginning

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Questions ???

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