Bayesian networks for mathematical models: Techniques for automatic construction and efficient inference

Expert knowledge in the form of mathematical models can be considered sufficient statistics of all prior experimentation in the domain, embodying generic or abstract knowledge of it. When used in a probabilistic framework, such models provide a sound foundation for data mining, inference, and decision making under uncertainty.We describe a methodology for encapsulating knowledge in the form of ordinary differential equations (ODEs) in dynamic Bayesian networks (DBNs). The resulting DBN framework can handle both data and model uncertainty in a principled manner, can be used for temporal data mining with noisy and missing data, and can be used to re-estimate model parameters automatically using data streams. A standard assumption when performing inference in DBNs is that time steps are fixed. Generally, the time step chosen is small enough to capture the dynamics of the most rapidly changing variable. This can result in DBNs having a natural time step that is very short, leading to inefficient inference; this is particularly an issue for DBNs derived from ODEs and for systems where the dynamics are not uniform over time.We propose an alternative to the fixed time step inference used in standard DBNs. In our algorithm, the DBN automatically adapts the time step lengths to suit the dynamics in each step. The resulting system allows us to efficiently infer probable values of hidden variables using multiple time series of evidence, some of which may be sparse, noisy or incomplete.We evaluate our approach with a DBN based on a variant of the van der Pol oscillator, and demonstrate an example where it gives more accurate results than the standard approach, but using only one tenth the number of time steps.We also apply our approach to a real-world example in critical care medicine. By incorporating knowledge in the form of an existing ODE model, we have built a DBN framework for efficiently predicting individualised patient responses using the available bedside and lab data.