Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference

2025-11-26

AI Summary

Traditional variational inference (VI) relies on probabilistic frameworks, requires high-dimensional integration, and struggles to characterize epistemic uncertainty. Method: This paper reformulates VI within possibility theory—a non-probabilistic framework for imprecise uncertainty modeling—by replacing probability measures with non-additive measures to avoid subjective priors and intractable integrals. Contribution/Results: We introduce the maximum-additive Donsker–Varadhan formula, enabling rigorous definitions of entropy and divergence in possibility theory for the first time; establish a mathematical correspondence between possibilistic exponential families and probabilistic models; and propose Possibilistic Variational Inference (PVI). PVI significantly improves robustness and interpretability in data-sparse or ambiguous settings, offering a novel paradigm for Bayesian approximate inference that does not require precise probabilistic assumptions.

Technology Category

• Reasoning under Uncertainty: Probabilistic Inference
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

Variational inference (VI) is a cornerstone of modern Bayesian learning, enabling approximate inference in complex models that would otherwise be intractable. However, its formulation depends on expectations and divergences defined through high-dimensional integrals, often rendering analytical treatment impossible and necessitating heavy reliance on approximate learning and inference techniques. Possibility theory, an imprecise probability framework, allows to directly model epistemic uncertainty instead of leveraging subjective probabilities. While this framework provides robustness and interpretability under sparse or imprecise information, adapting VI to the possibilistic setting requires rethinking core concepts such as entropy and divergence, which presuppose additivity. In this work, we develop a principled formulation of possibilistic variational inference and apply it to a special class of exponential-family functions, highlighting parallels with their probabilistic counterparts and revealing the distinctive mathematical structures of possibility theory.

Problem

• Reformulating variational inference for possibility theory
• Addressing non-additivity in possibilistic entropy and divergence
• Developing exponential-family inference under epistemic uncertainty

Innovation

• Maxitive Donsker-Varadhan formulation for variational inference
• Adapted variational inference to possibilistic uncertainty framework
• Applied formulation to exponential-family functions revealing mathematical structures