Generalised exponential families in category-theoretic probability

In applications of probability theory, exponential families are everywhere. In physics the Boltzmann distribution is the main example, and several machine learning algorithms can be seen as fitting parameters of an exponential family, including Boltzmann machines as well as many applications of neural networks. Exponential families also arise often in classical statistical inference.

Recently there is a lot of interest in category theoretic probability, which allows theorems about probability to be expressed in a more abstract way. Because of this, it seems important to understand exponential families from this new perspective.

In this talk I will show how (a known generalisation of) exponential families can be defined within the framework of Markov categories and show how this relates to this relates to conjugate priors and Bayesian inference. I will also discuss how the ideas can be generalised to stationary processes instead of i.i.d. data, which makes a connection to computational mechanics and epsilon machines.