Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Towards Tsallis Fully Probabilistic Design

Vyacheslav Kungurtsev, Giovanni Russo

TL;DR

By forming a fixed point iteration, this paper 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.

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.

Towards Tsallis Fully Probabilistic Design

TL;DR

By forming a fixed point iteration, this paper 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.

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.
Paper Structure (15 sections, 9 theorems, 39 equations, 1 algorithm)

This paper contains 15 sections, 9 theorems, 39 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Contributions
  3. Preliminaries
  4. Notation
  5. Tsallis Divergence: Definition and Basic Properties
  6. Fixed Point Theory
  7. Problem Statement
  8. Background on Fully Probabilistic Design (FPD)
  9. Tsallis Fully Probabilistic Design
  10. Fixed Point Iteration Scheme
  11. Theoretical Results
  12. Backwards Induction Expressions
  13. Fixed Point Convergence Guarantees
  14. Discussion and Conclusion
  15. Appendix

Key Result

Lemma 2.4

Let $p(\mathbf{v},\mathbf{z})$, $q(\mathbf{v},\mathbf{z})$ be two joint pdfs such that $q(\mathbf{v},\mathbf{z})=q_v(\mathbf{v})q_z(\mathbf{z})$ and $p(\mathbf{v},\mathbf{z})=p_{\vert}(\mathbf{v}\vert\mathbf{z})p_z(\mathbf{z})$. Then:

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2: Tsallis Divergence and Total Conditional Tsallis Divergence
  • Remark 2.3
  • Lemma 2.4
  • Remark 2.5
  • Lemma 2.6
  • Remark 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 5.1
  • ...and 8 more