A Dynamic Approach to Probabilistic Inference

In this paper we present a framework for dynamically constructing Bayesian net­ works. We introduce the notion of a back­ ground knowledge base of �chemata, . �hich is a collection of parameterized conditiOnal probability statements. These schemata explicitly separate the general knowledge of properties an individual may have from the specific knowledge of particular individuals that may have these properties. Knowledge of individuals can be combined with this background knowledge to create Bayesian networks, which can then be used in any propagation scheme. We discuss the theory and assumptions necessary for the implementation of dy­ namic Bayesian networks, and indicate where our approach may be useful. 1 Motivation Bayesian networks are used in AI applications to model uncertainty and perform inference. They are often used in expert systems [Andreasse n et al., 1987], and decision analysis[Schachter, 1988, Howard and Matheson, 1981], in which the network is engineered to perform a highly specialized analysis task. A Bayesian network often implicitly combines gen­ eral knowledge with specific knowledge. For exam­ ple, a Bayesian network with an arc as in Figure 1 refers to a specific individual (a house or a tree or dinner or whatever), exhibiting a somewhat gener­ alized property (fire causes smoke). Our dynamic approach is m�tivated by the '?b­ servation that a knowledge engmeer has expertise in a domain, but may not be able to anticipate t.he individuals in the model. By separating properties from individuals the knowledge engineer can write a knowledge base which is independent of the in�i­ viduals; the system user can tell the system which *This research is supported in part by NSERC grant #OGP0044121. individuals to consider, because she can make this observation at run time. The system user doesn't have to be an expert in the domain, to create an appropriate network. As an example, suppose we are using Bayesian networks to model probabilistically the response of several people to the sound of an alarm. Our ap­ proach allows the observation of any number of peo­ ple. The same knowledge about how people respond to alarms is used for each person. · There are two parts to our approach. First, we provide a collection of schemata, which are . param­ eterized, and can be used when necessary given .the details of the problem. In particular, the same piece of knowledge may be instantiated several times in a single dynamically created network. . . Second an automatic process bmlds a Bayesian network by combining the observation of individu­ als with the schemata. Thus, if we want to reason about a situation involving a fire alarm, given that three different people all hear the same alarm, this information, provided as evidence to our inference engine, causes the appropriate network to be cre­ ated. The Bayesian network constructed dynami­ cally can absorb evidence (conditioning) to P.ro­ vide posterior probabilities using any propagatiOn scheme[Pearl, 1988, Lauritzen and Spiegelhalter, 1988, Schachter, 1988]. We now proceed with a cursory introduction to Bayesian networks, followed by a presentation of our dynamic approach, giving some examples of how dy­ namic networks can be used. Finally, we draw some conclusions concerning the applicability of dynamic networks to particular domains. 2 Bayesian Networks A Bayesian network is a directed acyclic graph which represents in graphical form the joint probability dis­ tribution, and the statistical independence assump­ tions, for a set of random variables. A node in the graph represents a variable, and an arc indicates that the node at the head of the arc is directly dependent on, or conditioned by the node at Figure 1: A simple arc implicitly representing an individual. the tail. The collection of arcs directed to a variable give the independenc� assu�ption� for the. de�en­ dent variable. Associated with this collection IS a prior probability distribution, or contingency table, which quantifies the effects of observing events for the conditioning variables on the probability of the dependent variable. The graphical representation is used in various ways to calculate posterior joint probabilities. Pearl [Pearl, 1988] uses the arcs to propagate �ausal and diagnostic support values throughout a smgly­ connected network. Lauritzen and Spiegelhalter [Lauritzen and Spiegelhalter, 1988] perform evi­ dence absorption and propagation by constructing a triangulated graph based on the Bayesian net­ work that models the domain knowledge. Schachter [Schachter, 1988] uses the arcs to perform node re­ duction on the network.I Poole and Neufeld [Poole and Neufeld, 1989] implement an axiomatization of probability theory in Prolog which uses the arcs of the network to calculate probabilities using "reason­ ing by cases" with the conditioning variables. Our current implementation uses the work of Poole and Neufeld, but is not dependent on it. 3 Representation Issues Creating networks automatically raises several issues which we discuss in this section. First, we need to represent our background knowledge in a way which preserves a coherent joint distribution defined by Bayesian networks, and which also facilitates its use in arbitrary situations. Furthermore, there are sit­ uations in which using the background knowledge may lead to ambiguity, and our approach must also deal with this. Before we get into our discussion, a short section dealing with syntactic conventions will help make things clearer. 3.1 Some syntax In this section we clarify the distinction between our background knowledge and Bayesian networks. 1Schachter's influence diagrams contain decision and test nodes, as well as other devices not being considered in this paper. We are only looking at probabilistic nodes at present. 156 Figure 2: A simple instantiation of a schema. In our Bayesian networks, random variables are propositions written like ground Prolog terms. For example, the random variable fties{e127} cou.ld rep­ resent the proposition that individual e127 fhes. A schema is part of the background knowledge base, and describes qualitatively the direct depen­ dencies of a random variable. In this paper, a schema is stated declaratively in sans serif font. A schema can be defined constructively: a param­ eterized atom is written as a Prolog term with pa­ rameters capitalized. A schema is represented by the following: a1, ... , Bn -->b where the arrow � indicates that the parameter­ ized atom, b, on the left side is directly dependent on the parameterized atoms, ai, on the right. An instantiation of a parameter is the substitution of a parameter by a constant representing an indi­ vidual. We indicate this in our examples by listing the individuals in a set, as in {el27}. Instantiating a parameterized atom creates a proposition that can be used in a Bayesian network. A schema is instantiated when all parameters have been instantiated, and an instantiated schema be­ comes a part of the Bayesian network, indicating a directed arc from the instantiation. of each ai and the instantiation of b. For example, the schema: foo(Xa), bar(a) -foobar(X) with {b} instantiating the parameter X, creates the network2 shown in Figure 2. The schemata in the background knowledge base are isolated pieces of information, whereas a Bayesian network is a single entity. An interpreta­ tion is that a Bayesian network defines a joint distri­ bution, whereas the schemata in a knowledge base 2In our examples, Bayesian networks are pictured as encircled node labels connected by directed arrows, as in Figure 2. I