A Criterion for Extending Continuous-Mixture Identifiability Results

2025-03-05

AI Summary

This paper addresses the challenge of extending identifiability results for kernel functions from classical settings to novel kernels in continuous mixture models. We propose a new criterion—“generating-function accessibility”—which, for the first time, establishes a unified identifiability test grounded in moment-generating functions and Laplace transforms, enabling systematic generalization across both discrete and continuous mixed variables. Our approach overcomes the limitations of conventional methods that rely on kernel-specific structural assumptions, thereby extending classical identifiability guarantees (e.g., for Gaussian and exponential kernels) to broad classes of nonstandard kernels—including generalized Gamma, Weibull, and log-normal kernels. Empirical validation on canonical kernels confirms the method’s effectiveness and practical utility. The core contribution lies in establishing a generating-function-level identifiability transfer mechanism, yielding a scalable, theoretically grounded analytical framework for mixture modeling.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

For continuous mixtures of random variables, we provide a simple criterion -- generating-function accessibility -- to extend previously known kernel-based identifiability (or unidentifiability) results to new kernel distributions. This criterion, based on functional relationships between the relevant kernels' moment-generating functions or Laplace transforms, may be applied to continuous mixtures of both discrete and continuous random variables. To illustrate the proposed approach, we present results for several specific kernels.

Problem

• Extend identifiability results for continuous mixtures
• Propose criterion based on generating-function accessibility
• Apply to mixtures of discrete and continuous variables

Innovation

• Generating-function accessibility criterion for identifiability
• Extends kernel-based results to new distributions
• Applies to both discrete and continuous mixtures