Troubling how one’s thought traces fade. We were getting interested (or at least I was) about Shannon entropy, cohomology and characteristic classes a while ago, and Urs was even speculating about a route to a dynamics for probability distributions here via an umkehr map.

Now I see there’s an article

- Dalton A R Sakthivadivel,
*A Constraint Geometry for Inference and Integration*(arXiv:2203.08119)

which is looking to use gauge theory to think about statistical inference

We use the geometric framework describing gauge theories to enrich our understanding of the principle of maximum entropy, a variational method appearing in statistical inference and the analysis of stochastic dynamical systems. Using the connection on a principal G-bundle, the gradient flows found in the calculus of functional optimisation are grounded in a geometric picture of constraint functions interacting with the dynamics of the probabilistic degrees of freedom of the process. From this, we can describe the point of maximum entropy as parallel transport over the state space. An interpretation of splitting results in stochastic dynamical systems is also suggested. Beyond stochastic analysis, we indicate a collection of geometric structures surrounding energy-based inference.

Maybe some connections there.

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