  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.
