New preprint builds a compositional information geometry for probabilistic programs
FrnkNlsn · x · 2026-10-14
Frank Nielsen, Jacek Karwowski and Sam Staton released a new preprint, Compositional Information Geometry of Probabilistic Programs (arXiv:2610.17886).
- For a first-order probabilistic language with finite random variables, they build a contravariant lax monoidal functor that pulls back a degenerate Riemannian metric on Y to X for programs of type X→Y
- It specializes to the Fisher metric for standard statistical models and to ordinary pullback for deterministic programs
- The construction is justified two ways: as pullback of geometry from a universal space of programs over Y, and as a special case of a general pullback over weighted measures
- They establish a three-level inequality relating the program's score-velocity metric, the output-conditioned metric, and the Wasserstein-Fisher-Rao metric of the realized measures
- The theory extends to continuous random variables via diffeological quasi-Borel spaces
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