Stochastic Simulation of Bayesian Belief Networks

This paper examines the use of stochastic simulation of Bayesian belief networks as a method for computing the probabilities of values of variables. Specifically, it examines the use of a scheme described by Henrion, called logic sampling, and an extension to that scheme described by Pearl. The scheme devised by Pearl allows us to "clamp" any number of variables to given values and to conduct stochastic simulation on the resulting network. We have found that this algorithm, in certain networks, leads to much slower than expected convergence to the true posterior probability. This behavior is a result of the tendency for local areas in the graph to become fixed through many stochastic iterations. The length of this non-convergence can be made arbitrarily long by strengthening the dependency between two nodes. This paper describes the use of graph modification. By modifying a belief network through the use of pruning, arc reversal, and node reduction, it may be possible to convert the network to a form that is computationally more efficient for stochastic simulation. 1 . Introduction The graphical representation of probabilistic relationships between events has been the subject of considerable research. In the field of artificial intelligence, numerous systems have used a directed graph to represent probabilistic relationships between events [Duda 76, Weiss 78]. A particular probabilistic graphical representation has been independently defined and explored by several researchers. In addition to being called belief networks [Pearl 86], they have been termed causa/ nets [Good 61 ], probabilistic cause-effect models [Rousseau 68], probabilistic causal networks [Cooper 84], and influence diagrams [Howard 84, Shachter 86]. Graphical representation of probabilistic relationships allows for the efficient representation of all the existing dependencies between variables. It is only necessary to consider the known dependencies among the variables in a domain, rather than assuming that all variables are dependent on all other variables. This leads to a significant decrease in the number of probabilities needed to define the outcome space, and improved computational efficiency. Although probabilistic graphical representations often allow efficient probabilistic inference, some inference problems involving particular topological classes of belief networks have been resistant to any efficient algorithmic solution [Pearl 86]. Multiply-connected networks, the most general class of such problems, belong to a class of difficult problems which have are resistant to any general, efficient solutions, and have been shown to belong to the class of NP-hard problems [Cooper 87]. All known exact algorithms for performing inference over multiply-connected belief networks are exponential in the size of the belief network. 2. Stochastic Simulation of Bayesian Belief Networks Because exact solution of multiply connected networks is ekponentially complex in the worst cases, the development of stochastic simulation techniques to generate probabilities for variables in the network has been an area of considerable research interest. One such method, presented by Henrion [Henrion 86], is logic sampling. In a Bayesian belief network where variables are represented by nodes, simulation is begun with the nodes that have no parent nodes. A value is assigned to each such node based on its prior probability of occurrence. For example, if the probabilities for the values of a node are P(HIGH) = 0.4, P(MEDIUM) = 0.4, and P(LOW) = 0.2, then a value for this node can be simulated by generating a random number between 0 and 1.0. For numbers between 0 and 0.4, the node is set to the value HIGH; for numbers between 0.4 and 0.8, the node is set to MEDIUM; and for numbers above 0.8, the node is set to LOW. The children of these nodes can be similarly assigned values based on the conditional probabilities relating them to their parents. This process is continued until all the nodes in the network have been simulated. After many simulations, the probability of a value for a given variable can be approximated by dividing the number of times a variable is assigned a given value by the number of simulations performed. The proximity of this approximation to the true probability of a value for a node can be determined by calculating the statistical variance.