Monotonicity in Bayesian Networks

For many real-life Bayesian networks, common knowledge dictates that the output established for the main variable of interest increases with higher values for the observable variables. We define two concepts of monotonicity to capture this type of knowledge. We say that a network is isotone in distribution if the probability distribution computed for the output variable given specific observations is stochastically dominated by any such distribution given higher-ordered observations; a network is isotone in mode if a probability distribution given higher observations has a higher mode. We show that establishing whether a network exhibits any of these properties of monotonicity is coNPPP-complete in general, and remains coNP-complete for polytrees. We present an approximate algorithm for deciding whether a network is monotone in distribution and illustrate its application to a real-life network in oncology.Comment: Appears in Proceedings of the Twentieth Conference on Uncertainty in Artificial Intelligence (UAI2004

Verifying monotonicity in Bayesian networks with domain experts

In many real problem domains, the main variable of interest behaves monotonically in the observable variables, in the sense that higher values for the variable of interest become more likely with higher-ordered observations. This type of knowledge appears to arise naturally during knowledge elicitation, without explicit prompting. The monotonicity properties declared by experts, however, may not correspond to the mathematical concept of monotonicity in Bayesian networks. We present a method for verifying, with the help of the experts, whether or not a network exhibits the implied properties of monotonicity. We illustrate the application of our method for a real Bayesian network in veterinary science.