Maximum‐likelihood maximum‐entropy constrained probability density function estimation for prediction of rare events

This work addresses the problem of estimating complete probability density functions (PDFs) from historical process data that are incomplete (lack information on rare events), in the framework of Bayesian networks. In particular, this article presents a method of estimating the probabilities of events for which historical process data have no record. The rare‐event prediction problem becomes more difficult and interesting, when an accurate first‐principles model of the process is not available. To address this problem, a novel method of estimating complete multivariate PDFs is proposed. This method uses the maximum entropy and maximum likelihood principles. It is tested on mathematical and process examples, and the application and satisfactory performance of the method in risk assessment and fault detection are shown. Also, the proposed method is compared with a few copula methods and a nonparametric kernel method, in terms of performance, flexibility, interpretability, and rate of convergence.