Bayesian improved cross entropy method with categorical mixture models

TL;DR

This paper introduces a Bayesian improved cross entropy method using categorical mixture models for efficient rare event estimation in networks, incorporating Bayesian inference and EM algorithms to improve accuracy and prevent overfitting.

Contribution

The paper develops a novel BiCE method with categorical mixture models, employing Bayesian MAP updates and a generalized EM algorithm, enhancing rare event estimation accuracy.

Findings

Outperforms standard iCE in accuracy and efficiency

Effectively captures dependence among network components

Provides unbiased importance sampling distribution

Abstract

We employ the Bayesian improved cross entropy (BiCE) method for rare event estimation in static networks and choose the categorical mixture as the parametric family to capture the dependence among network components. At each iteration of the BiCE method, the mixture parameters are updated through the weighted maximum a posteriori (MAP) estimate, which mitigates the overfitting issue of the standard improved cross entropy (iCE) method through a novel balanced prior, and we propose a generalized version of the expectation-maximization (EM) algorithm to approximate this weighted MAP estimate. The resulting importance sampling distribution is proved to be unbiased. For choosing a proper number of components $K$ in the mixture, we compute the Bayesian information criterion (BIC) of each candidate $K$ as a by-product of the generalized EM algorithm. The performance of the proposed method is investigated through a simple illustration, a benchmark study, and a practical application. In all these numerical examples, the BiCE method results in an efficient and accurate estimator that significantly outperforms the standard iCE method and the BiCE method with the independent categorical distribution.