Unified Causality Analysis Based on the Degrees of Freedom

TL;DR

This paper presents a unified method capable of identifying fundamental causal relationships between pairs of systems, whether deterministic or stochastic, and also uncovers hidden common causes beyond the observed variables.

Abstract

Temporally evolving systems are typically modeled by dynamic equations. A key challenge in accurate modeling is understanding the causal relationships between subsystems, as well as identifying the presence and influence of unobserved hidden drivers on the observed dynamics. This paper presents a unified method capable of identifying fundamental causal relationships between pairs of systems, whether deterministic or stochastic. Notably, the method also uncovers hidden common causes beyond the observed variables. By analyzing the degrees of freedom in the system, our approach provides a more comprehensive understanding of both causal influence and hidden confounders. This unified framework is validated through theoretical models and simulations, demonstrating its robustness and potential for broader application.

Figures (7)

• Figure 1: Observation of the $z$ subsystem: we apply different numbers of constraints using different bin widths (window widths). Top row: "deterministic" case; bottom row: "stochastic" case (for parameter assignments, see text).
• Figure 2: Observation of the $x$ subsystem: different numbers of constraints are applied with varying window widths. The left panel shows the "deterministic" case, and the right panel corresponds to the "stochastic" case (for parameter assignments, see text). The two stabilized lines indicate that this subsystem has 2 degrees of freedom (counting in pairs).
• Figure 3: Observation of the $x$-$z$ subsystem: different numbers of constraints are applied with varying window widths, and we study the width of the observed $x$ distribution. The left panel shows the "deterministic" case, and the right panel shows the "stochastic" case (for parameter assignments, see text).
• Figure 4: Observation of the $x$-$y$ subsystem: different numbers of constraints are applied using different bin widths (window widths), and we study the width of the observed $y$ distribution. The left panel corresponds to the "deterministic" case, and the right panel corresponds to the "stochastic" case (for parameter assignments, see text).
• Figure 5: Chicken and egg data, normalized.

Theorems & Definitions (3)

• Definition 1
• Definition 2
• Definition 3