  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.
