Probabilistic Inference in In uence Diagrams

This paper studies the relationship between probabilistic inference in Bayesian networks and evaluation of in uence diagrams We clearly identify and separate from other computations probabilistic inference subproblems that must be solved in order to evaluate an in uence diagram This work leads to a new method for evaluating in uence diagrams where arbitrary Bayesian network inference algorithms can be used for probabilistic inference We argue that if the VE algorithm Zhang and Poole Zhang and Poole is used for probabilistic inference the new method is more e cient than the best previous methods Shenoy Jensen et al If a more e cient probabilistic inference algorithm such as the VEC algorithm Zhang and Poole or some approximation algorithm is used the new method can be even more e cient Keyword Decision analysis in uence diagrams Bayesian networks inference Introduction Traditionally decision analysis is carried out by using decision trees Rai a Decision trees grow exponentially in size as the number of decisions and observations increases A more compact framework called in uence diagrams was introduced by Howard and Matheson in In addition to being more compact than decision trees in uence diagrams are also more intuitive and reveal more problem structures They have enabled researchers to solve large decision problems that are beyond the capabilities of decision trees Bayesian networks Pearl are a knowledge representation framework for reasoning under uncertainty that is widely by AI researchers There is a rich collection of exact and approximate algorithms for inference in Bayesian networks An in uence diagram is an acyclic graph with three types of nodes random nodes decision nodes and a single value node while a Bayesian network is an acyclic graph consisting of only random nodes Evaluation of an in uence diagram refers to the process of nding optimal decision rules for its decision nodes Probabilistic inference in a Bayesian network refers to the process of computing the posterior probabilities of nodes of interest given observations It is widely recognized that in uence diagram evaluation and inference in Bayesian networks are closely related Shenoy Shachter and Peot Jensen et al This paper clari es the relationship by clearly identify and separate from other computations probabilistic inference subproblems that must be solved in order to evaluate an in uence diagram This work leads to a new method for evaluating in uence diagrams where arbitrary Bayesian network inference algorithms can be used for probabilistic inference Our work is built upon earlier e orts to utilize Bayesian network inference algo rithms in in uence diagram evaluation Shenoy has extended the idea behind a popular Bayesian network inference algorithm called junction tree propagation and proposed a fusion algorithm for evaluating in uence diagrams The algorithm was later re ned by Jensen et al in terms of strong junction trees propagation Shachter and Peot have shown how an in uence diagram can be transformed into a Bayesian network such that optimal decision rules can be computed by posing a sequence of queries to the Bayesian network The queries generated by Shachter and Peot s method can be answered by us ing arbitrary Bayesian network inference algorithms This means that progresses in Bayesian network inference can be readily used to bene t in uence diagram evalua tion Shenoy s fusion algorithm and strong junction tree propagation on the other hand are mixtures of probabilistic inference expected utility calculation and opti mization New advances in Bayesian network inference cannot be directly incorpo rated However Shachter and Peot s method is less e cient than Shenoy s fusion algorithm and strong junction tree propagation The new method proposed in this paper shares the advantage of Shachter and Peot s method When the VE algorithm Zhang and Poole Zhang and Poole is used for probabilistic inference the new method is also more e cient than Also known as cluster tree propagation and clique tree propagation Shenoy s fusion algorithm and strong junction tree propagation If a more e cient probabilistic inference algorithm such as the VEC algorithm Zhang and Poole or some approximation algorithm is used the new method can be even more e cient We shall begin with de nitions related to in uence diagrams Section and some technical preparations Section The foundation for our new method will be lead in Section and details will be worked out in Section The method will then be illustrated by using an example Section and compared to previous methods Section Conclusions will be presented in Section In uence diagrams In uence diagrams Howard and Matheson are acyclic graphs with three types of nodes decision nodes random nodes and value nodes The decision nodes rep resent decisions to make The random nodes represent random quantities relevant to the decisions Arcs into random nodes indicate probabilistic dependence and the dependence of a random node c upon its parents c is characterized by a conditional probability P cj c The value nodes represents components of the utility function y Each value node v is associated with a value function a real valued function fv v of the parents v of v The total utility function is the summation of all the value functions Arcs into decision nodes are indicate information availability and are some times called informational arcs In uence diagrams are required to satisfy several constraints Firstly value nodes cannot have children Secondly in uence diagrams must be regular in the sense that there must be a directed path that contains all the decision nodes The last decision node on such a path will be referred to as the tail decision node Thirdly they must be no forgetting in the sense that a decision node and its parents be parents to all subsequent decision nodes The rationale behind the no forgetting constraint is that information available now should also be available later if the decision maker does not forget Value networks refer to in uence diagrams that do not contain decision nodes Bayesian networks Pearl are in uence diagrams that consists of only random nodes In the following the terms nodes and variables will be used interchangeably Optimal polices We shall use x to denote the frame of variable x i e the set of possible values of x For a set X of variables X stands for the Cartesian product Q x X x Let d dk be all the decision nodes in an in uence diagram N A decision rule for a decision node di is a mapping i di di A policy is a list of decision rules k consisting of one rule for each decision node In the original de nition of in uence diagrams Howard and Matheson there is only one value node We allow multiple value nodes here so that separability in the utility function can be represented As a convention we say that there is only one possible policy the vacuous policy for value networks Given a decision rule i de ne a conditional probability P i dij di by P i dij di if i di di otherwise Since k we shall sometimes write P i dij di as P dij di Given a policy the decision nodes can be viewed as random nodes Ignoring the value nodes one can view N as a Bayesian network Let C and D be the set of random nodes and decision nodes respectively The Bayesian network represents the following joint probability P C D Y