Page 1 Module 10 Reasoning with Uncertainty - Probabilistic reasoning Version 2 CSE IIT, Kharagpur Page 2 Module 10 Reasoning with Uncertainty - Probabilistic reasoning Version 2 CSE IIT, Kharagpur Lesson 28 Bayes Networks Version 2 CSE IIT, Kharagpur Page 3 Module 10 Reasoning with Uncertainty - Probabilistic reasoning Version 2 CSE IIT, Kharagpur Lesson 28 Bayes Networks Version 2 CSE IIT, Kharagpur 10.5 Bayesian Networks 10.5.1 Representation and Syntax Bayes nets (BN) (also referred to as Probabilistic Graphical Models and Bayesian Belief Networks) are directed acyclic graphs (DAGs) where each node represents a random variable. The intuitive meaning of an arrow from a parent to a child is that the parent directly influences the child. These influences are quantified by conditional probabilities. BNs are graphical representations of joint distributions. The BN for the medical expert system mentioned previously represents a joint distribution over 8 binary random variables {A,T,E,L,S,B,D,X}. Conditional Probability Tables Each node in a Bayesian net has an associated conditional probability table or CPT. (Assume all random variables have only a finite number of possible values). This gives the probability values for the random variable at the node conditional on values for its parents. Here is a part of one of the CPTs from the medical expert system network. If a node has no parents, then the CPT reduces to a table giving the marginal distribution on that random variable. Version 2 CSE IIT, Kharagpur Page 4 Module 10 Reasoning with Uncertainty - Probabilistic reasoning Version 2 CSE IIT, Kharagpur Lesson 28 Bayes Networks Version 2 CSE IIT, Kharagpur 10.5 Bayesian Networks 10.5.1 Representation and Syntax Bayes nets (BN) (also referred to as Probabilistic Graphical Models and Bayesian Belief Networks) are directed acyclic graphs (DAGs) where each node represents a random variable. The intuitive meaning of an arrow from a parent to a child is that the parent directly influences the child. These influences are quantified by conditional probabilities. BNs are graphical representations of joint distributions. The BN for the medical expert system mentioned previously represents a joint distribution over 8 binary random variables {A,T,E,L,S,B,D,X}. Conditional Probability Tables Each node in a Bayesian net has an associated conditional probability table or CPT. (Assume all random variables have only a finite number of possible values). This gives the probability values for the random variable at the node conditional on values for its parents. Here is a part of one of the CPTs from the medical expert system network. If a node has no parents, then the CPT reduces to a table giving the marginal distribution on that random variable. Version 2 CSE IIT, Kharagpur Consider another example, in which all nodes are binary, i.e., have two possible values, which we will denote by T (true) and F (false). We see that the event "grass is wet" (W=true) has two possible causes: either the water sprinker is on (S=true) or it is raining (R=true). The strength of this relationship is shown in the table. For example, we see that Pr(W=true | S=true, R=false) = 0.9 (second row), and hence, Pr(W=false | S=true, R=false) = 1 - 0.9 = 0.1, since each row must sum to one. Since the C node has no parents, its CPT specifies the prior probability that it is cloudy (in this case, 0.5). (Think of C as representing the season: if it is a cloudy season, it is less likely that the sprinkler is on and more likely that the rain is on.) 10.5.2 Semantics of Bayesian Networks The simplest conditional independence relationship encoded in a Bayesian network can be stated as follows: a node is independent of its ancestors given its parents, where the ancestor/parent relationship is with respect to some fixed topological ordering of the nodes. In the sprinkler example above, by the chain rule of probability, the joint probability of all the nodes in the graph above is Version 2 CSE IIT, Kharagpur Page 5 Module 10 Reasoning with Uncertainty - Probabilistic reasoning Version 2 CSE IIT, Kharagpur Lesson 28 Bayes Networks Version 2 CSE IIT, Kharagpur 10.5 Bayesian Networks 10.5.1 Representation and Syntax Bayes nets (BN) (also referred to as Probabilistic Graphical Models and Bayesian Belief Networks) are directed acyclic graphs (DAGs) where each node represents a random variable. The intuitive meaning of an arrow from a parent to a child is that the parent directly influences the child. These influences are quantified by conditional probabilities. BNs are graphical representations of joint distributions. The BN for the medical expert system mentioned previously represents a joint distribution over 8 binary random variables {A,T,E,L,S,B,D,X}. Conditional Probability Tables Each node in a Bayesian net has an associated conditional probability table or CPT. (Assume all random variables have only a finite number of possible values). This gives the probability values for the random variable at the node conditional on values for its parents. Here is a part of one of the CPTs from the medical expert system network. If a node has no parents, then the CPT reduces to a table giving the marginal distribution on that random variable. Version 2 CSE IIT, Kharagpur Consider another example, in which all nodes are binary, i.e., have two possible values, which we will denote by T (true) and F (false). We see that the event "grass is wet" (W=true) has two possible causes: either the water sprinker is on (S=true) or it is raining (R=true). The strength of this relationship is shown in the table. For example, we see that Pr(W=true | S=true, R=false) = 0.9 (second row), and hence, Pr(W=false | S=true, R=false) = 1 - 0.9 = 0.1, since each row must sum to one. Since the C node has no parents, its CPT specifies the prior probability that it is cloudy (in this case, 0.5). (Think of C as representing the season: if it is a cloudy season, it is less likely that the sprinkler is on and more likely that the rain is on.) 10.5.2 Semantics of Bayesian Networks The simplest conditional independence relationship encoded in a Bayesian network can be stated as follows: a node is independent of its ancestors given its parents, where the ancestor/parent relationship is with respect to some fixed topological ordering of the nodes. In the sprinkler example above, by the chain rule of probability, the joint probability of all the nodes in the graph above is Version 2 CSE IIT, Kharagpur P(C, S, R, W) = P(C) * P(S|C) * P(R|C,S) * P(W|C,S,R) By using conditional independence relationships, we can rewrite this as ause R is independent of S given its arent C, and the last term because W is independent of C given its parents S and R. We from a parent to a child is that the parent directly fluences the child. The direction of this influence is often taken to represent casual n into a product of conditional probabilities sing repeated applications of the product rule. ditionally independent of all P(C, S, R, W) = P(C) * P(S|C) * P(R|C) * P(W|S,R) where we were allowed to simplify the third term bec p can see that the conditional independence relationships allow us to represent the joint more compactly. Here the savings are minimal, but in general, if we had n binary nodes, the full joint would require O(2^n) space to represent, but the factored form would require O(n 2^k) space to represent, where k is the maximum fan-in of a node. And fewer parameters makes learning easier. The intuitive meaning of an arrow in influence. The conditional probabilities give the strength of causal influence. A 0 or 1 in a CPT represents a deterministic influence. 10.5.2.1 Decomposing Joint Distributions A joint distribution can always be broken dow u We can order the variables however we like: 10.5.2.2 Conditional Independence in Bayes Net A Bayes net represents the assumption that each node is con its non-descendants given its parents. So for example, Version 2 CSE IIT, KharagpurRead More

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