Advances in Decision Analysis: Model Building with Belief Networks and Influence Diagrams

Belief networks and influence diagrams use directed graphs to represent models for probabilistic reasoning and decision making under uncertainty. They have proven to be effective at facilitating communication with decision makers and with computers. Many of the important relationships among uncertainties, decisions, and values can be captured in the structure of these diagrams, explicitly revealing irrelevance and the flow of information. We explore a variety of examples illustrating some of these basic structures, along with an algorithm that efficiently analyzes their model structure. We also show how algorithms based on these structures can be used to resolve inference queries and determine the optimal policies for decisions. CONTENTS Probabilistic Models Analyzing Probabilistic Models Decision Models Analyzing Decision Models Advances: Belief Nets and Influence Diagrams Page 2 of 42 Ch 10 060520 V09 We have all learned how to translate models, as we prefer to think of them, into arcane representations that our computers can understand, or to simplify away key subtleties for the benefit of clients or students. Thus it has been an immense pleasure to work with graphical models where the representation is natural for the novice, convenient for computation, and yet powerful enough to convey difficult concepts among analysts and researchers. The graphical representations of belief networks and influence diagrams enable us to capture important relationships at the structural level of the graph where it easiest for people to see them and for algorithms to exploit them. Although the diagrams lend themselves to communication, there remains the challenge of synthesis, and building graphical models is still a challenging art. This chapter presents many examples of model structures within these representations and some of their most important properties. It is designed for students and practitioners with a basic understanding of decision analysis. An example illustrates the value of a clear representation. For many years the accepted model for contract bidding (Friedman, 1956) corresponded to the network shown in Figure 10.1a, where “&” represents background information. Buried in the mathematics was an incorrect assumption that if we know our bid then the bids of our competitors provides us no new information about the value of the contract. It was only after analyzing bidding patterns that had cost oil companies many extra millions of dollars in Gulf Coast leases that Capen, Clapp, & Campbell (1971) discovered the mistake and presented the model corresponding to the network shown in Figure 10.1b. In their model, the value of the contract renders our bid and the bids of our competitors irrelevant to each other, assuming no collusion. As a result we have the “winner’s curse,” because observing that we won the bid makes it more likely that our bid is in Advances: Belief Nets and Influence Diagrams Page 3 of 42 Ch 10 060520 V09 error. In hindsight and with the insights from graphical models, it is disturbing that the error went undetected for so many years. & Competitor Bid 1 Our Bid Contract Value Competitor Bid 2 a) b) & Competitor Bid 1 Our Bid Contract Value Competitor Bid 2 Figure 10.1. Different models for contract bidding The use of a directed graph factorization of a joint distribution is usually credited to (Wright, 1921, 1934) and (Good, 1961a, 1961b), although it was first fully described by (Howard & Matheson, 1984; Miller, Merkofer, Howard, Matheson, & Rice, 1976). It came to computer science and statistics attention through (Kim & Pearl, 1983; Lauritzen & Spiegelhalter, 1988; Pearl, 1988). Since then, there have many significant developments in model building and analysis, particularly with the use of undirected graphs, but our focus here is on the basic models in directed graphs. In the next section we present examples of probabilistic models represented by belief networks, followed by a section featuring algorithms to explore the belief network structure and use simple directed graphical transformations to analyze inference queries. The final two sections present examples of decision models represented by influence diagrams, and algorithms and directed transformations to analyze them. Advances: Belief Nets and Influence Diagrams Page 4 of 42 Ch 10 060520 V09 Probabilistic Models In this section we consider examples of probabilistic models represented by belief networks. Such networks are known by many names, including Bayesian networks, relevance diagrams, causal probabilistic networks, and recursive causal models. The structure of the graph captures irrelevance information about the underlying probabilistic model through deterministic variables and the arcs that could be present but are missing. The data associated with the nodes in the graph provides a complete probabilistic description of the uncertainties in the model and allows an updating of that description as variables are observed. The structure of a belief network B = (U, A, F | &) consists of undirected nodes U, directed arcs A between those nodes, and a subset F of the nodes that are determined (functionally) by other nodes. Corresponding to each node j is an uncertain variable Xj, and XJ refers to the variables corresponding to the set of nodes J. The background state of information & represents the perspective of the decision maker (DM) at a point in time. It captures her beliefs, biases, and life experiences in the context of the current situation. Consider the belief network shown in Figure 10.2a, representing a contest in which the prize is determined by two coin spins. Spin 1 and Spin 2 are oval probabilistic nodes and Prize is a double oval deterministic node. If both spins are observed then we will know the prize with certainty, but if they are not then we might be uncertain about Prize. The background state of information & is represented by a dark oval node, which might be labeled with the date and/or the DM’s name. In Figure 10.2a, Spin 1 and Spin 2 direct arcs toward Prize, so they are its parents and it is their child. Similarly, in Figure 10.2b if Spin 1 and Prize are known, then they are the parents and their child, Spin 2, is also known. We denote the parents of node j by Pa(j) Advances: Belief Nets and Influence Diagrams Page 5 of 42 Ch 10 060520 V09 and following the familial analogy, the set of nodes along directed paths from node j are called its descendants, De(j). Spin 1 Spin Spin Spin b) a) 2 1 2