Use of Kullback–Leibler divergence for forgetting

Abstract

Non-symmetric Kullback-Leibler divergence (KLD) measures proximity of probability density functions (pdfs). Bernardo (Ann. Stat. 1979; 7(3):686-690) had shown its unique role in approximation of pdfs. The order of the KLD arguments is also implied by his methodological result. Functional approximation of estimation and stabilized forgetting, serving for tracking of slowly varying parameters, use the reversed order. This choice has the pragmatic motivation: recursive estimator often approximates the parametric model by a member of exponential family (EF) as it maps prior pdfs from the set of conjugate pdfs (CEF) back to the CEF. Approximations based on the KLD with the reversed order of arguments preserves this property. In the paper, the approximation performed within the CEF but with the proper order of arguments of the KLD is advocated. It is applied to the parameter tracking and performance improvements are demonstrated. This practical result is of importance for adaptive systems and opens a way for improving the functional approximation.

Online Prediction Under Model Uncertainty via Dynamic Model Averaging: Application to a Cold Rolling Mill

“…Although the resulting model does not specify the transition matrix of the Markov chain explicitly, Smith and Miller (1986) argued that this is not a defect of the model, since the data provide information about π t −1| t −1, k and π t | t −1, k , but no additional information about Q . Exponential forgetting has been used for updating discrete probabilities in a different two-hypothesis context by Kárný and Andrýsek (2009).…”

Recursive Identification of Time-Varying Hammerstein Systems With Matrix Forgetting

“…At the Bayesian level, a sort of forgetting arises through combining the posterior pdf with its flattened alternative. The combination strategies prominently involve the nonsymmetric Kullback-Leibler divergence (KLD) [8] with different properties depending on the order of the KLD arguments [9]. There is rich literature on the adaptation of a single forgetting factor causing the information about all of the system parameters to be uniformly discounted [10]- [13].…”