On Properties of Doeblin Coefficients

Doeblin coefficients are a classical tool to study the ergodicity 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. Specifically, we present various new structural and geometric properties of Doeblin coefficients. Then, by establishing an extremal coupling characterization, we show that Doeblin coefficients generalize 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 also demonstrate that Doeblin coefficients exhibit contraction properties over Bayesian networks similar to other canonical contraction coefficients. Finally, we discuss how Doeblin coefficients can be used to define a new rule for fusion of probability mass functions.