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Doeblin Coefficients and Related Measures

Anuran Makur, Japneet Singh

TL;DR

The paper studies Doeblin coefficients as a natural, multi-way generalization of total variation that captures contraction through channels and Bayesian networks. It develops a rich geometric and structural theory, including a maximal coupling characterization, a new max-Doeblin coefficient with a corresponding minimal coupling, and multi-distribution DeGroot distances, enabling multi-way divergence measures. It further shows contraction properties over Bayesian networks, presents SDPI-type results in the Doeblin framework, and applies these ideas to PMF fusion via an optimization-based, coupling-driven approach. The results unify Doeblin-coefficient theory with canonical contraction-coefficient ideas, offering practical tools for multi-source decision problems and probabilistic fusion with rigorous probabilistic guarantees.

Abstract

Doeblin coefficients are a classical tool for analyzing the ergodicity and exponential convergence rates of Markov chains. Propelled by recent works on contraction coefficients of strong data processing inequalities, we investigate whether Doeblin coefficients also exhibit some of the notable properties of canonical contraction coefficients. In this paper, we present several new structural and geometric properties of Doeblin coefficients. Specifically, we show that Doeblin coefficients form a multi-way divergence, exhibit tensorization, and possess an extremal trace characterization. We then show that they also have extremal coupling and simultaneously maximal coupling characterizations. By leveraging these characterizations, we demonstrate that Doeblin coefficients act as a nice generalization of the well-known total variation (TV) distance to a multi-way divergence, enabling us to measure the "distance" between multiple distributions rather than just two. We then prove that Doeblin coefficients exhibit contraction properties over Bayesian networks similar to other canonical contraction coefficients. We additionally derive some other results and discuss an application of Doeblin coefficients to distribution fusion. Finally, in a complementary vein, we introduce and discuss three new quantities: max-Doeblin coefficient, max-DeGroot distance, and min-DeGroot distance. The max-Doeblin coefficient shares a connection with the concept of maximal leakage in information security; we explore its properties and provide a coupling characterization. On the other hand, the max-DeGroot and min-DeGroot measures extend the concept of DeGroot distance to multiple distributions.

Doeblin Coefficients and Related Measures

TL;DR

The paper studies Doeblin coefficients as a natural, multi-way generalization of total variation that captures contraction through channels and Bayesian networks. It develops a rich geometric and structural theory, including a maximal coupling characterization, a new max-Doeblin coefficient with a corresponding minimal coupling, and multi-distribution DeGroot distances, enabling multi-way divergence measures. It further shows contraction properties over Bayesian networks, presents SDPI-type results in the Doeblin framework, and applies these ideas to PMF fusion via an optimization-based, coupling-driven approach. The results unify Doeblin-coefficient theory with canonical contraction-coefficient ideas, offering practical tools for multi-source decision problems and probabilistic fusion with rigorous probabilistic guarantees.

Abstract

Doeblin coefficients are a classical tool for analyzing the ergodicity and exponential convergence rates of Markov chains. Propelled by recent works on contraction coefficients of strong data processing inequalities, we investigate whether Doeblin coefficients also exhibit some of the notable properties of canonical contraction coefficients. In this paper, we present several new structural and geometric properties of Doeblin coefficients. Specifically, we show that Doeblin coefficients form a multi-way divergence, exhibit tensorization, and possess an extremal trace characterization. We then show that they also have extremal coupling and simultaneously maximal coupling characterizations. By leveraging these characterizations, we demonstrate that Doeblin coefficients act as a nice generalization of the well-known total variation (TV) distance to a multi-way divergence, enabling us to measure the "distance" between multiple distributions rather than just two. We then prove that Doeblin coefficients exhibit contraction properties over Bayesian networks similar to other canonical contraction coefficients. We additionally derive some other results and discuss an application of Doeblin coefficients to distribution fusion. Finally, in a complementary vein, we introduce and discuss three new quantities: max-Doeblin coefficient, max-DeGroot distance, and min-DeGroot distance. The max-Doeblin coefficient shares a connection with the concept of maximal leakage in information security; we explore its properties and provide a coupling characterization. On the other hand, the max-DeGroot and min-DeGroot measures extend the concept of DeGroot distance to multiple distributions.
Paper Structure (39 sections, 17 theorems, 180 equations, 1 figure)

This paper contains 39 sections, 17 theorems, 180 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Related Literature on Doeblin Coefficients
  3. Main Contributions
  4. Outline
  5. Notation
  6. Main Results and Discussion
  7. Doeblin Coefficient and Its Maximal Coupling Characterization
  8. Properties of Doeblin Coefficients
  9. Maximal Coupling Characterization for Doeblin Coefficient
  10. Max-Doeblin Coefficient and Its Minimal Coupling Characterization
  11. Properties of Max-Doeblin Coefficient
  12. Minimal Coupling Characterization of Max-Doeblin Coefficient
  13. DeGroot Distance
  14. Min-DeGroot Distance
  15. Max-DeGroot Distance
  16. ...and 24 more sections

Key Result

Theorem 1

Let $W = [P_1^{\mathrm{T}} \, \dots \, P_n^{\mathrm{T}}]^{\mathrm{T}} \in {\mathbb{R}}^{n \times m}_{\mathsf{sto}}$ be a channel formed by stacking the PMFs $P_{1}, \ldots, P_{n} \in {\mathcal{P}}_m$. Then, the Doeblin coefficient $\tau(W)$ of $W$ satisfies the following properties:

Figures (1)

  • Figure 1: Illustration of a Bayesian network depicting a source node denoted by $X$, a sink node $U$, and a subset $V$ consisting of nodes enclosed within dotted lines.

Theorems & Definitions (23)

  • Definition 1: Doeblin Coefficient
  • Theorem 1: Properties of Doeblin Coefficients
  • Theorem 2: Maximal Coupling for Doeblin Coefficient
  • Definition 2: Max-Doeblin Coefficient
  • Theorem 3: Properties of Max-Doeblin Coefficient
  • Theorem 4: Minimal Coupling for Max-Doeblin Coefficient
  • Proposition 1: Minimal Coupling for Max-Doeblin Coefficient when $n=3$
  • Proposition 2: Simultaneous Coupling for Doeblin and Max-Doeblin Coefficients
  • Theorem 5: Simultaneously Maximal Coupling
  • Theorem 6: Doeblin Coefficients in Bayesian Networks
  • ...and 13 more