[Paper Review] Causal Modeling with Probabilistic Simulation Models

Duligur Ibeling|arXiv (Cornell University)|Jul 30, 2018

Bayesian Modeling and Causal Inference8 references3 citations

TL;DR

This paper extends causal simulation models to probabilistic settings by defining a formal semantics for counterfactual reasoning over probabilistic programs. It introduces a logic for reasoning about probabilities of conditionals, proves soundness and completeness of the axiomatization, and establishes NP-completeness of the satisfiability problem for linear inequalities over probabilities.

ABSTRACT

Recent authors have proposed analyzing conditional reasoning through a notion of intervention on a simulation program, and have found a sound and complete axiomatization of the logic of conditionals in this setting. Here we extend this setting to the case of probabilistic simulation models. We give a natural definition of probability on formulas of the conditional language, allowing for the expression of counterfactuals, and prove foundational results about this definition. We also find an axiomatization for reasoning about linear inequalities involving probabilities in this setting. We prove soundness, completeness, and NP-completeness of the satisfiability problem for this logic.

Motivation & Objective

• To formalize counterfactual reasoning in probabilistic simulation models, extending prior work on non-probabilistic causal simulations.
• To define a semantics for probabilities over conditional formulas in a way that aligns with standard probability theory.
• To develop a logical framework for reasoning about linear inequalities involving probabilities of conditionals in this setting.
• To prove soundness and completeness of the resulting logic, and to analyze the computational complexity of its satisfiability problem.
• To lay foundational groundwork for integrating probabilistic reasoning into simulation-based causal models, especially for cognitive modeling and Bayesian logic programming.

Proposed method

• Models probabilistic simulation programs as probabilistic Turing machines with access to a random bit tape, ensuring all underlying computation is deterministic.
• Defines interventions on simulation models as functions that fix tape square values to counterfactual values, preserving the program’s execution flow.
• Introduces a formal language ℒ for expressing probabilistic conditionals, including formulas like ℙ(φ) ≤ c, where φ is a conditional statement.
• Establishes that probabilities over formulas can be interpreted as genuine probabilities by showing consistency with Kolmogorov axioms.
• Constructs a finite, polynomial-sized model ℳφ,C,D for any formula φ, using rational approximations of probabilities derived from program branches.
• Proves NP-completeness of the satisfiability problem by reducing it to a system of linear inequalities with rational coefficients of bounded size.

Experimental results

Research questions

• RQ1 How can counterfactual reasoning be formally extended to probabilistic simulation models?
• RQ2 What is a sound and complete axiomatization for reasoning about probabilities of conditionals in this framework?
• RQ3 How does the computational complexity of reasoning about probabilistic conditionals compare to the non-probabilistic case?
• RQ4 In what subclasses of simulation models do classical logical principles like cautious monotonicity hold?
• RQ5 How can first-order and open-world reasoning be integrated into this simulation-based causal modeling framework?

Key findings

• The paper provides a sound and complete axiomatization for reasoning about linear inequalities involving probabilities of conditionals in probabilistic simulation models.
• The satisfiability problem for this logic is proven to be NP-complete, establishing a precise complexity bound for reasoning in the framework.
• Probabilities assigned to formulas in the language are shown to satisfy the axioms of probability theory, validating their interpretation as genuine probabilities.
• A finite model ℳφ,C,D of polynomial size exists for any formula φ, ensuring the logic is effectively representable and decidable in NP.
• The construction of models involves rational approximations of probabilities derived from program branches, with coefficients bounded in length by O(|φ|³).
• The framework supports conditioning on events via restricting to runs satisfying a query, aligning with the conditional simulation approach in cognitive modeling.