On Discovery of Local Independence over Continuous Variables via Neural Contextual Decomposition
Inwoo Hwang, Yunhyeok Kwak, Yeon-Ji Song, Byoung-Tak Zhang, Sanghack Lee
TL;DR
The paper tackles the problem of discovering local causal structure when variables are continuous by introducing context-set specific independence (CSSI), a generalization of CSI/PCI that accommodates continuous conditioned variables. It develops a canonical representation via contextual decomposition (CD) and a partition-based view of CSSIs, then implements Neural Contextual Decomposition (NCD) that learns a partition indicator variable (PIV) to identify multiple CSSIs. NCD models conditional distributions $p(y|\mathbf{x}, z)$ and $p(z|\mathbf{x})$ with neural networks and optimizes a marginalized likelihood objective using Gumbel-Softmax for differentiable partition learning. Empirical results on synthetic data and Spriteworld demonstrate accurate recovery of ground-truth CSSIs and reveal the learned context boundaries, highlighting the method’s potential to enable fine-grained causal analysis in systems with continuous variables. The work offers a principled framework for continuous local independence and a scalable neural approach to uncovering complex context-specific causal structure.
Abstract
Conditional independence provides a way to understand causal relationships among the variables of interest. An underlying system may exhibit more fine-grained causal relationships especially between a variable and its parents, which will be called the local independence relationships. One of the most widely studied local relationships is Context-Specific Independence (CSI), which holds in a specific assignment of conditioned variables. However, its applicability is often limited since it does not allow continuous variables: data conditioned to the specific value of a continuous variable contains few instances, if not none, making it infeasible to test independence. In this work, we define and characterize the local independence relationship that holds in a specific set of joint assignments of parental variables, which we call context-set specific independence (CSSI). We then provide a canonical representation of CSSI and prove its fundamental properties. Based on our theoretical findings, we cast the problem of discovering multiple CSSI relationships in a system as finding a partition of the joint outcome space. Finally, we propose a novel method, coined neural contextual decomposition (NCD), which learns such partition by imposing each set to induce CSSI via modeling a conditional distribution. We empirically demonstrate that the proposed method successfully discovers the ground truth local independence relationships in both synthetic dataset and complex system reflecting the real-world physical dynamics.