Bayesian Updates Compose Optically — Toby St. Clere Smithe (2020). arXiv:2006.01631 (v2, PDF).
Shows that Bayesian inversion composes like the backward pass of a lens: inversions are state-dependent morphisms, organised as Grothendieck lenses for a state-indexed category, equivalently as optics for an action on presheaves. The main theorem is that the Bayesian inverse of a composite channel is the lens composite of the inverses, so exact inversion embeds channels functorially into Bayesian lenses.
Sources: the paper, arXiv:2006.01631v2, checked against the arXiv listing. Index: Papers.
Key definitions and results
- Definitions 2.2–2.5: copy-delete categories, Bayesian inversion, almost-equality
- Definition 3.1: state-indexed categories; Definition 3.4: Grothendieck lenses
- Definition 4.3, Propositions 4.4–4.5: BayesLens as optics and as Grothendieck lenses
- Theorem 5.2, Corollary 5.3: Bayesian updates compose optically
Concept notes
Bayesian Lens, Bayesian Inversion, Grothendieck Construction, Optic