Develop a stronger Chentsov-type characterization for program-generated families

Prove a Chentsov-type characterization theorem for suitable classes of probabilistic-program-generated families, or characterize the classes of program subfamilies closed under operations for which such an invariance theorem holds.

Background

The paper gives a characterization of the score-velocity metric under relatively weak assumptions and compares this with classical characterizations of Fisher information. It notes that Chentsov-style invariance results may leave additional metrics possible on restricted families, and identifies the development of a stronger theorem for finite, infinite, smooth, compact, or otherwise specified program families as an unresolved direction.

References

For smaller families the theorem leaves open the possibility of there being a metric that is invariant on a smaller (but still somehow interesting) subset.

— Compositional Information Geometry of Probabilistic Programs  (2610.17886 - Karwowski et al., 13 Oct 2026) in Section 8, subsection "Chentsov's theorems for program-generated families"