Graded String Diagrams for Imprecise Probability and Causal Intervention

TL;DR

This paper develops a framework of graded string diagrams within symmetric monoidal categories, enabling modular axiomatization and representation of probabilistic and causal models with imprecise probabilities.

Contribution

It introduces graded string diagrams and a modular axiomatic approach, applying it to model imprecise probability and causal interventions.

Findings

Framework for graded symmetric monoidal categories

Modular axiomatization of graded theories

Representation of probabilistic programs with string diagrams

Abstract

We introduce string diagrams for graded symmetric monoidal categories. Our approach includes a definition of graded monoidal theory and the corresponding freely generated syntactic category. Also, we show how an axiomatic presentation for the graded theory may be modularly obtained from one for the grading theory and one for the base category. The Para construction on monoidal actegories is a motivating example for our framework. As a case study, we show how to axiomatise a variant of the graded category ImP, recently introduced by Liell-Cock and Staton to model imprecise probability. This culminates in a representation, as string diagrams with grading wires, of programs with primitives for nondeterministic and probabilistic choices and conditioning.