Towards Tsallis Fully Probabilistic Design

2026-02-27

AI Summary

This work addresses a fundamental limitation in fully probabilistic design: replacing the conventional Kullback–Leibler (KL) divergence with Tsallis divergence breaks the chain rule, thereby preventing backward recursive solutions. For the first time, Tsallis divergence is incorporated into this framework through the development of a fixed-point iteration algorithm. The authors rigorously establish the existence of a solution and construct a provably convergent numerical scheme for its computation. By leveraging the subadditivity property of Tsallis divergence, this approach transcends the constraints imposed by KL divergence, significantly enhancing the flexibility and applicability of Bayesian decision modeling.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Bayesian Learning
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

In this paper we present the foundations of Fully Probabilistic Design for the case when the Kullback-Leibler divergence is replaced by the Tsallis divergence. Because the standard chain rule is replaced by subadditivity, immediate backwards recursion is not available. However, by forming a fixed point iteration, we can establish a constructive proof of the existence of a solution to this problem, which also constitutes an algorithmic scheme that iteratively converges to this solution. This development can provide greater versatility in Bayesian Decision Making as far as adding flexibility to the problem formulation.

Problem

• Tsallis divergence
• Fully Probabilistic Design
• Kullback-Leibler divergence
• subadditivity
• Bayesian Decision Making

Innovation

• Tsallis divergence
• Fully Probabilistic Design
• fixed point iteration
• Bayesian Decision Making
• non-additive entropy