Automated reasoning has been receiving much interest from a number of fields, including philosophy, cognitive science, and computer science. In this chapter, we consider the particular interest of computer science in automated reasoning over the last few decades, and then focus our attention on probabilistic reasoning using Bayesian networks, which is the main subject of this book.

Automated reasoning

The interest in automated reasoning within computer science dates back to the very early days of artificial intelligence (AI), when much work had been initiated for developing computer programs for solving problems that require a high degree of intelligence. Indeed, an influential proposal for building automated reasoning systems was extended by John McCarthy shortly after the term “artificial intelligence” was coined [McCarthy, 1959]. This proposal, sketched in Figure 1.1, calls for a system with two components: a knowledge base, which encodes what we know about the world, and a reasoner (inference engine), which acts on the knowledge base to answer queries of interest. For example, the knowledge base may encode what we know about the theory of sets in mathematics, and the reasoner may be used to prove various theorems about this domain.

McCarthy's proposal was actually more specific than what is suggested by Figure 1.1, as he called for expressing the knowledge base using statements in a suitable logic, and for using logical deduction in realizing the reasoning engine; see Figure 1.2. McCarthy's proposal can then be viewed as having two distinct and orthogonal elements.

We address in this chapter a number of problems that arise in real-world applications, showing how each can be solved by modeling and reasoning with Bayesian networks.

Introduction

We consider a number of real-world applications in this chapter drawn from the domains of diagnosis, reliability, genetics, channel coding, and commonsense reasoning. For each one of these applications, we state a specific reasoning problem that can be addressed by posing a formal query with respect to a corresponding Bayesian network. We discuss the process of constructing the required network and then identify the specific queries that need to be applied.

There are at least four general types of queries that can be posed with respect to a Bayesian network. Which type of query to use in a specific situation is not always trivial and some of the queries are guaranteed to be equivalent under certain conditions. We define these query types formally in Section 5.2 and then discuss them and their relationships in more detail when we go over the various applications in Section 5.3.

The construction of a Bayesian network involves three major steps. First, we must decide on the set of relevant variables and their possible values. Next, we must build the network structure by connecting the variables into a DAG. Finally, we must define the CPT for each network variable. The last step is the quantitative part of this construction process and can be the most involved in certain situations.

As we discussed in the previous chapter, social choice theory is nonstrategic; it takes the preferences of the agents as given, and investigates ways in which they can be aggregated. But of course those preferences are usually not known. What you have, instead, is that the various agents declare their preferences, which they may do truthfully or not. Assuming the agents are self interested, in general they will not reveal their true preferences. Since as a designer you wish to find an optimal outcome with respect to the agents' true preferences (e.g., electing a leader that truly reflects the agents' preferences), optimizing with respect to the declared preferences will not in general achieve the objective.

Introduction

Mechanism design is a strategic version of social choice theory, which adds the assumption that agents will behave so as to maximize their individual payoffs. For example, in an election agents may not vote their true preference.

Example: strategic voting

Consider again our babysitting example. This time, in addition to Will, Liam, and Vic you must also babysit their devious new friend, Ray. Again, you invite each child to select their favorite among the three activities—going to the video arcade (a), playing basketball (b), and going for a leisurely car ride (c). As before, you announce that you will select the activity with the highest number of votes, breaking ties alphabetically.

Probability theory provides a formal framework for the discussion of chance or uncertainty. This appendix reviews some key concepts of the theory and establishes notation. However, it glosses over some details (e.g., pertaining to measure theory). Therefore, the interested reader is encouraged to consult a textbook on the topic for a more comprehensive picture.

Probabilistic models

A probabilistic model is defined as a tuple (Ω, F, P), where:

• Ω is the sample space, also called the event space;

• F is a σ-algebra over Ω; that is, F ⊆ 2Ω and is closed under intersection and countable union; and

• P : F ↦ [0, 1] is the probability density function (PDF).

Intuitively, the sample space is a set of things that can happen in the world according to our model. For example, in a model of a six-sided die, we might have Ω = {1, 2, 3, 4, 5, 6}. The σ-field F is a collection of measurable events. F is required because some outcomes in Ω may not be measurable; thus, we must define our probability density function P over F rather than over Ω. However, in many cases, such as the six-sided die example, all outcomes are measurable. In those cases we can equate F with 2Ω and view the probability space as the pair (Ω, P) and P as P : 2Ω ↦ [0, 1]. We assume this in the following.

Agents communicate; this is one of the defining characteristics of a multiagent system. In traditional linguistic analysis, the communication is taken to have a certain form (syntax), to carry a certain meaning (semantics), and to be influenced by various circumstances of the communication (pragmatics). As we shall see, a closer look at communication adds to the complexity of the story. We can distinguish between purely informational theories of communication and motivational ones. In informational communication, agents simply inform each other of different facts. The theories of belief change, introduced in Chapter 14, look at ways in which beliefs change in the face of new information—depending on whether the beliefs are logical or probabilistic, consistent with prior beliefs or not. In this chapter we broaden the discussion and consider motivational theories of communication, involving agents with individual motivations and possible courses of actions.

We divide the discussion into three parts. The first concerns cheap talk and describes a situation in which self-motivated agents can engage in costless communication before taking action. As we see, in some situations this talk influences future behavior, and in some it does not. Cheap talk can be viewed as “doing by talking”; in contrast, signaling games can be viewed as “talking by doing.” In signaling games an agent can take actions that, by virtue of the underlying incentives, communicate to the other agent something new.

The following is not intended as an introduction to classical logic, but rather as a review of the concepts and a setting of notation. We start with propositional calculus and then move to first-order logic. (We do the latter for completeness, but in fact first-order logic plays almost no role in this book.)

Propositional calculus

Syntax

Given a set P of propositional symbols, the set of sentences in the propositional calculus is the smallest set ℒ containing P such that if φ, ψ ∈ ℒ then also ¬φ ∈ ℒ and ∈ ∧ ψ ℒ. Other connectives such as ∨, →, and ≡ can be defined in terms of ∧ and ¬.

Semantics

A propositional interpretation (or a model) is a set M ⊂ P, the subset of true primitive propositions. The satisfaction relation ⊧ between models and sentences is defined recursively as follows.

• For any p ∈ P, M ⊧ p iff p ∈ M.

• M ⊧ φ ∧ ψ iff M ⊧ φ and M ⊧ ψ.

• M ⊧ ¬φ iff it is not the case that M ⊧ φ.

We overload the ⊧ symbol. First, it is used to denote validity; ⊧ φ means that φ is true in all propositional models. Second, it is used to denote entailment; φ ⊧ ψ means that any model that satisfies φ also satisfies ψ.

In this chapter we will go beyond the normal and extensive forms by considering a variety of richer game representations. These further representations are important because the normal and extensive forms are not always suitable for modeling large or realistic game-theoretic settings.

First, we may be interested in games that are not finite and that therefore cannot be represented in normal or extensive form. For example, we may want to consider what happens when a simple normal-form game such as the Prisoner's Dilemma is repeated infinitely. We might want to consider a game played by an uncountably infinite set of agents. Or we may want to use an interval of the real numbers as each player's action space.

Second, both of the representations we have studied so far presume that agents have perfect knowledge of everyone's payoffs. This seems like a poor model of many realistic situations, where, for example, agents might have private information that affects their own payoffs and other agents might have only probabilistic information about each others' private information. An elaboration like this can have a big impact, because one agent's actions can depend on what he knows about another agent's payoffs.

Finally, as the numbers of players and actions in a game grow—even if they remain finite—games can quickly become far too large to reason about or even to write down using the representations we have studied so far.

In this chapter and the next we discuss cooperative situations in which agents collaborate to achieve a common goal. This goal can be viewed as shared between the agents or, alternatively, as the goal of a central designer who is designing the various agents. Of course, if such a designer exists, a natural question is why it matters that there are multiple agents; they can be viewed merely as end sensors and effectors for executing the plan devised by the designer. However, there exist situations in which a problem needs to be solved in a distributed fashion, either because a central controller is not feasible or because one wants to make good use of the distributed resources. A good example is provided by sensor networks. Such networks consist of multiple processing units, each with local sensor capabilities, limited processing power, limited power supply, and limited communication bandwidth. Despite these limitations, these networks aim to provide some global service. Figure 1.1 shows an example of a fielded sensor network used for monitoring environmental quantities like humidity, temperature and pressure in an office environment. Each sensor can monitor only its local area and, similarly, can communicate only with other sensors in its local vicinity. The question is what algorithm the individual sensors should run so that the center can still piece together a reliable global picture.

In the previous chapter we looked at distributed ways of meeting global constraints. Here we up the ante; we ask how agents can, in a distributed fashion, optimize a global objective function. Specifically, we consider four families of techniques and associated sample problems. They are, in order:

• distributed dynamic programming (as applied to path-planning problems);

• distributed solutions to Markov Decision Problems (MDPs);

• optimization algorithms with an economic flavor (as applied to matching and scheduling problems); and

• coordination via social laws and conventions, and the example of traffic rules.

Distributed dynamic programming for path planning

Like graph coloring, path planning constitutes another common abstract problemsolving framework. A path-planning problem consists of a weighted directed graph with a set of n nodes N, directed links L, a weight function w : L ↦ ℝ+, and two nodes s, t ∈ N. The goal is to find a directed path from s to t having minimal total weight. More generally, we consider a set of goal nodes T ⊂ N, and are interested in the shortest path from s to any of the goal nodes t ∈ T.

This abstract framework applies in many domains. Certainly it applies when there is some concrete network at hand (e.g., a transportation or telecommunication network). But it also applies in more roundabout ways.

The discussion of strategies and solution concepts in Chapter 3 largely ignored issues of computation. We start by asking the most basic question: How hard is it to compute the Nash equilibria of a game? The answer turns out to be quite subtle, and to depend on the class of games being considered.

We have already seen how to compute the Nash equilibria of simple games. These calculations were deceptively easy, partly because there were only two players and partly because each player had only two actions. In this chapter we discuss several different classes of games, starting with the simple two-player, zero-sum normal-form game. Dropping only the zero-sum restriction yields a problem of different complexity—while it is generally believed that any algorithm that guarantees a solution must have an exponential worst case complexity, it is also believed that a proof to this effect may not emerge for some time. We also consider procedures for n-player games. In each case, we describe how to formulate the problem, the algorithm (or algorithms) commonly used to solve them, and the complexity of the problem. While we focus on the problem of finding a sample Nash equilibrium, we will briefly discuss the problem of finding all Nash equilibria and finding equilibria with specific properties. Along the way we also discuss the computation of other game-theoretic solution concepts: maxmin and minmax strategies, strategies that survive iterated removal of dominated strategies, and correlated equilibria.

In the preceding chapters we adopted what might be called the “agent perspective”: we asked what an agent believes or wants, and how an agent should or would act in a given situation. We now adopt a complementary, “designer perspective”: we ask what rules should be put in place by the authority (the “designer”) orchestrating a set of agents. In this chapter this will take us away from game theory, but before too long (in the next two chapters) it will bring us right back to it.

Introduction

A simple example of the designer perspective is voting. How should a central authority pool the preferences of different agents so as to best reflect the wishes of the population as a whole? It turns out that voting, the kind familiar from our political and other institutions, is only a special case of the general class of social choice problems. Social choice is a motivational but nonstrategic theory—agents have preferences, but do not try to camouflage them in order to manipulate the outcome (of voting, for example) to their personal advantage. This problem is thus analogous to the problem of belief fusion that we present in Section 14.2.1, which is also nonstrategic; here, however, we examine the problem of aggregating preferences rather than beliefs.