Reference. Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion

Dylan Braithwaite, Jules Hedges · arXiv · 2022 · lenses probabilistic · arXiv
We generalise an existing construction of Bayesian Lenses to admit lenses between pairs of objects where the backwards object is dependent on states on the forwards object (interpreted as probability distributions). This gives a natural setting for studying stochastic maps with Bayesian inverses restricted to the points supported by a given prior. In order to state this formally we develop a proposed definition by Fritz of a support object in a Markov category and show that these give rise to a section into the category of dependent Bayesian lenses encoding a more canonical notion of Bayesian inversion.

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Cite as @braithwaite-2022-dependent (helia, typst) · \cite{braithwaite-2022-dependent} (LaTeX)
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bibtex · 8 lines
@misc{braithwaite-2022-dependent,
  author = {Dylan Braithwaite and Jules Hedges},
  title = {Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion},
  year = {2022},
  month = {9},
  eprint = {2209.14728},
  archiveprefix = {arXiv}
}
hayagriva YAML (typst)
yaml · 9 lines
braithwaite-2022-dependent:
  type: misc
  title: 'Dependent Bayesian Lenses: Categories of Bidirectional Markov Kernels with Canonical Bayesian Inversion'
  author:
  - Braithwaite, Dylan
  - Hedges, Jules
  date: 2022-09
  serial-number:
    arxiv: '2209.14728'
Download: braithwaite-2022-dependent.bib · braithwaite-2022-dependent.yml
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Cites 20 works (4 here)
With notes (4)

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We lift the standard equivalence between fibrations and indexed categories to an equivalence between monoidal fibrations and monoidal indexed categories, namely lax monoidal pseudofunctors to the 2-category of categories. Furthermore, we investigate the relation between this ‘global’ monoidal version where the total category is monoidal and the fibration strictly preserves the structure, and a ‘fibrewise’ one where the fibres are monoidal and the reindexing functors strongly preserve the structure, first hinted by Shulman. In particular, when the domain is cocartesian monoidal, we show how lax monoidal structures on a pseudofunctor to Cat bijectively correspond to lifts of the pseudofunctor to MonCat. Finally, we give some examples where this correspondence appears, spanning from the fundamental and family fibrations to network models and systems.
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External (16)
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