Define a probability composition rule for propagating uncertainty along multi-step regulatory pathways

Develop a general-purpose probability composition operator ∘P for the probabilistically enriched path category G_prob in the PC-GRN framework, specifying how to compute the distribution D_{f∘g} for any composite path (morphism) from the distributions D_f and D_g associated with its constituent edges, in order to rigorously propagate uncertainty along multi-step regulatory pathways derived from the influence graph.

Background

Within the PC-GRN framework, each regulatory interaction (edge) in the influence graph induces a morphism in a probabilistically enriched path category G_prob. Single-edge morphisms carry a probability distribution D_f derived from posterior kinetic parameter distributions of the corresponding Bayesian Typed Petri Net (BTPN) arc. For longer paths, the framework requires a rule ∘P to propagate and compose these distributions, but a general-purpose operator is not yet formalized.

Earlier in the formalism (Layer 3), the paper specifies that the distribution for a longer path should be obtained by propagating uncertainty via a composition rule ∘P, with a pointer to Section 5.3 for discussion. In Section 5.3, the authors explicitly identify defining a general-purpose ∘P as an open problem and outline possible directions, including Monte Carlo composition, analytical composition for special distribution families, and learning a composition rule as a meta-model.

References

While defining a general-purpose rule is a challenging open problem, several promising avenues for its definition can be explored:

— Modeling GRNs with a Probabilistic Categorical Framework  (2508.13208 - Jia et al., 16 Aug 2025) in Section 5.3 (Potential Impact and Avenues for Future Research), Theoretical Development

This Bayesian updating is the mechanism behind the endogenous risk shift: clearing milestones resolves risk and reprices the remaining option portfolio. The framework treats this updating as conceptual; specifying a functional form for $p_{s+1 \mid s\;\text{cleared}$ would risk overspecification for a framework paper and is deferred to future work.

— Firm Valuation When AI Shapes the Business Model: A Milestone-Based Real-Options Framework for the AI Valuation Uncertainty Problem  (2609.24181 - Kurz et al., 21 Sep 2026) in Section 3, subsection “Lifecycle and empirical design”

Propagation of uncertainty through the composition laws remains an open question.

— Rotational Trajectory Synopsis Groups  (2610.01691 - Chirikjian et al., 1 Oct 2026) in Conclusion