Inference on diagrams in the category of Markov kernels (Extended abstract)
Résumé
We propose a Bayesian inference perspective on how to make inferences on hierarchies of heterogeneous signals motivated by modeling agents with a heterogeneous representation of their environment that must communicate their beliefs in a collaborative exploration task. Solving this problem leads us to consider diagrams, more precisely partial orders, in the category of probability kernels and the resolution of the sub-problem of making inference for those presheaves. Inference on such presheaves complements the more restrictive theory of inference for graphical models and factor graphs, allowing us to make the junction between Bayesian inference and novel databases of signals. Cellular sheaves have been recently proposed as a way to model the exchange of heterogeneous signals through a graph structured network and are the building block for deep learning architectures to make decisions over such signals (sheaf neural networks). Cellular sheaves are shaped on cell complexes and the associated posets are not all possible posets. The structures we consider depart from cellular sheaves as we consider a more general setting where the hierarchy can be modeled by any poset. The message passing algorithm we propose, which directly generalizes Belief propagation, is different from the one considered in sheaf neural networks, and contrary to sheaf neural networks, its nonlinearity is given by the minimization of entropy; in other words, it is given by the optimization problem we solve (maximizing entropy) and not a priori as a design of the network, contrary to sheaf neural networks.
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