From distributional data in finance to the Bregman-Wasserstein divergence
Geometric structures on spaces of probability distributions arise naturally in theory and application. In the first part of the talk, we present some distributional data-sets (i.e., each data point is a probability distribution) motivated by stochastic portfolio theory and analyze them using Aitchison geometry and Wasserstein geodesic PCA. The latter is based on quadratic optimal transport, but the resulting 2-Wasserstein metric is just one of many possible divergences based on optimal transport. In the second part of the talk, we introduce the Bregman-Wasserstein divergence where the underlying cost function is a Bregman divergence. We show that this divergence is tractable and induces an elegant geometric structure on the space of probability distributions. We end with a discussion of some potential applications. Based on joint works with Steven Campbell (arXiv:2211.02990) and Cale Rankin (arXiv:2302.05833).