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

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.

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”

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

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

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

Independence Statement / Markovian Assumption

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)

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

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.

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

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

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

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)

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

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

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

Questions ???