I haven't yet read through them thoroughly, but these four papers by Oliver Richardson are pattern-matching to me as potentially very exciting theoretical work.

tl;dr: probabilistic dependency graphs (PDGs) are directed graphical models designed to be able to capture inconsistent beliefs (paper 1). The definition of inconsistency is a natural one which allows us to, for example, reframe the concept of "minimizing training loss" as "minimizing inconsistency" (paper 2). They provide an algorithm for inference in PDGs (paper 3) and an algorithm for learning via locally minimizing inconsistency which unifies several other algorithms (like the EM algorithm, message-passing, and generative adversarial training) (paper 4).

Oliver is an old friend of mine (which is how I found out about these papers) and a final-year PhD student at Cornell under Joe Halpern.