Computational Intelligence, Volume 16, Number 3, 2000 TEMPORAL REASONING AND BAYESIAN NETWORKS Ahmed Y. Tawfik University of Prince Edward Island, Charlottetown, PE, C1A 4P3 Canada Eric M. Neufeld University of Saskatchewan, Saskatoon, SK, S7N 5A9 Canada This work examines important issues in probabilistic temporal representation and reasoning using Bayesian networks (also known as belief networks). The representation proposed here utilizes temporal (or dynamic) probabilities to represent facts, events, and the effects of events. The architecture of a belief network may change with time to indicate a different causal context. Probability variations with time capture temporal properties such as persistence and causation. They also capture event interaction, and when the interaction between events follows known models such as the competing risks model, the additive model, or the dominating event model, the net effect of many interacting events on the temporal probabilities can be calculated efficiently. This representation of reasoning also exploits the notion of temporal degeneration of relevance due to information obsolescence to improve the efficiency. Key words: temporal representation and reasoning, uncertain reasoning, Bayesian (belief) networks, models of interaction. 1. INTRODUCTION Recent efforts to introduce temporality into Bayesian networks have resulted in a variety of networks intended primarily for applications such as planning, diagnosis, forecasting, and scheduling. Dynamic nets (Dagum, Galper, and Horvitz 1992) use an instantiation of the network for each time point, with the different instantiations linked by edges representing persistence and causation. Temporal Bayes nets (Dean and Kanazawa 1989) use survival functions to represent persistence. The underlying time model is discrete and each time point corresponds to a copy of the network. Arcs linking two copies propagate the effects of previous states and observations. Net- works of dates (Berzuini 1990) represent a departure from the multiple instantiations approach because each temporal duration is represented by a node. Berzuini associates a probability density with each temporal random variable to represent continuous time. Time nets (Kanazawa 1992) define a network model that uses continuous time and extends the “networks of dates” by introducing a representation for facts (or fluents). The dHugin time-sliced Bayesian nets (Kjaerulff 1995) are based on the multiple- instantiation approach (each time slice corresponds to a copy of the network) similar to temporal Bayes nets and dynamic nets but the reasoning is based on a dynamic version of Hugin and a smoothing operator is used to approximate the effect of temporally distant occurrences. The kappa calculus approximation of probability functions and the representation of persistence through suppressors are two features introduced in action networks (Darwiche and Goldszmidt 1994). Action networks use different instantiations of the original network for different time points. A time-sliced Bayes nets generation algorithm (Ngo, Haddawy, and Helwig 1995) optimizes the network to answer a query efficiently. Despite these efforts, there is no consensus on several issues such as when to dupli- cate the network, how to represent instantaneous effects, what conclusions can be made regarding the time interval between two instantiations, and how to represent continuous Address correspondence to A. Y. Tawfik at the University of Prince Edward Island, Charlottetown, PE, C1A 4P3 Canada; e-mail: atawfik@upei.ca c  2000 Blackwell Publishers, 350 Main St., Malden, MA 02148, USA, and 108 Cowley Road, Oxford, OX4 1JF, UK. 349