The data drawn from biological, economic, and social systems are often
confounded due to the presence of unmeasured variables. Prior work in causal
discovery has focused on discrete search procedures for selecting acyclic
directed mixed graphs (ADMGs), specifically ancestral ADMGs, that encode
ordinary conditional independence constraints among the observed variables of
the system. However, confounded systems also exhibit more general equality
restrictions that cannot be represented via these graphs, placing a limit on
the kinds of structures that can be learned using ancestral ADMGs. In this
work, we derive differentiable algebraic constraints that fully characterize
the space of ancestral ADMGs, as well as more general classes of ADMGs, arid
ADMGs and bow-free ADMGs, that capture all equality restrictions on the
observed variables. We use these constraints to cast causal discovery as a
continuous optimization problem and design differentiable procedures to find
the best fitting ADMG when the data comes from a confounded linear system of
equations with correlated errors. We demonstrate the efficacy of our method
through simulations and application to a protein expression dataset. Code
implementing our methods is open-source and publicly available at
this https URL and will be incorporated into the Ananke
package.

通过不可测量变量存在的混淆系统推导不可追踪混杂图（ADMGs）的状态，从而将因果发现视为连续优化问题，并设计不同的可微过程来发现线性方程组的最佳拟合 ADMG 结构。