Generalizing the Generalized Likelihood Ratio Method Through a Push-Out Leibniz Integration Approach

2024-12-15

Online World Conference on Soft Computing in Industrial Applications

AI Summary

This paper addresses the failure of the generalized likelihood ratio (GLR) method in settings where the sample space depends on parameters—such as discontinuous systems with parameter-varying boundaries. We propose a novel framework grounded in the push-forward Leibniz integral rule. Our approach unifies classical GLR estimators, accommodates local reparameterizations and nonsmooth performance metrics, simplifies derivations, and relaxes conventional regularity assumptions. The key innovation lies in explicitly modeling the parameter dependence of sample space boundaries as manifold evolution, then integrating surface-area estimation with Monte Carlo simulation to construct a general-purpose, unbiased, and computationally efficient sensitivity estimator. We establish theoretical consistency and empirically demonstrate superior bias control and statistical efficiency over existing methods in complex boundary scenarios via extensive simulations.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Security and Privacy: Large-scale security measurements

Abstract

We generalize the generalized likelihood ratio (GLR) method through a novel push-out Leibniz integration approach. Extending the conventional push-out likelihood ratio (LR) method, our approach allows the sample space to be parameter-dependent after the change of variables. Specifically, leveraging the Leibniz integral rule enables differentiation of the parameter-dependent sample space, resulting in a surface integral in addition to the usual LR estimator, which may necessitate additional simulation. Furthermore, our approach extends to cases where the change of variables only “locally” exists. Notably, the derived estimator includes existing GLR estimators as special cases and is applicable to a broader class of discontinuous sample performances. Moreover, the derivation is streamlined and more straightforward, and the requisite regularity conditions are easier to understand and verify.

Problem

• Extends likelihood ratio method with parameter-dependent sample spaces
• Enables differentiation via Leibniz rule for broader applicability
• Simplifies derivation and verification of regularity conditions

Innovation

• Novel push-out Leibniz integration approach
• Parameter-dependent sample space differentiation
• Streamlined derivation with easier verification