  In information geometry, a strictly convex and smooth function induces a
dually flat Hessian manifold equipped with a pair of dual Bregman divergences,
hereby termed a Bregman manifold. Two common types of such Bregman manifolds
met in statistics are (1) the exponential family manifolds induced by the
cumulant functions of regular exponential families, and (2) the mixture family
manifolds induced by the Shannon negentropies of statistical mixture families
with prescribed linearly independent mixture components. However, the
differential entropy of a mixture of continuous probability densities sharing
the same support is hitherto not known in closed form making implementation of
mixture family manifolds in practice difficult. In this work, we report an
exception: The family of mixtures of two prescribed and distinct Cauchy
distributions. We exemplify the explicit construction of a dually flat manifold
induced by the differential negentropy for this very particular setting. This
construction allows one to use the geometric toolbox of Bregman algorithms, and
to obtain closed-form formula (albeit being large) for the Kullback-Leibler
divergence and the Jensen-Shannon divergence between two mixtures of two
prescribed Cauchy components.
