Alpha/Beta Divergences and Tweedie Models

We describe the underlying probabilistic interpretation of alpha and beta divergences. We first show that beta divergences are inherently tied to Tweedie distributions, a particular type of exponential family, known as exponential dispersion models. Starting from the variance function of a Tweedie model, we outline how to get alpha and beta divergences as special cases of Csisz\'ar's $f$ and Bregman divergences. This result directly generalizes the well-known relationship between the Gaussian distribution and least squares estimation to Tweedie models and beta divergence minimization.