Consider the Monge-Kantorovich optimal transport problem where the cost function is given by a Bregman divergence. The associated transport cost, termed the Bregman-Wasserstein divergence here, presents a natural asymmetric extension of the (squared) 2-Wasserstein metric and has recently found applications in statistics and machine learning. On the other hand, Bregman divergence is a fundamental concept in information geometry and induces a dually flat geometry on the underlying manifold. Using the Bregman-Wasserstein divergence, we lift this dualistic geometry to the space of probability measures, yielding an extension of Otto’s weak Riemannian structure of the Wasserstein space to statistical manifolds. We do this by generalizing Lott’s formal geometric computations for the Wasserstein space. In particular, we define generalized displacement interpolations which are compatible with the Bregman geometry, and construct conjugate primal and dual connections on the space of distributions. We also discuss some potential applications. Ongoing joint work with Cale Rankin.