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.