Determining potential probability distributions with a given causal graph is vital for causality studies. To bypass the difficulty in characterizing latent variables in a Bayesian network, the nested Markov model provides an elegant algebraic approach by listing exactly all the equality constraints on the observed variables. However, this algebraically motivated causal model comprises distributions outside Bayesian networks, and its physical interpretation remains vague. In this work, we inspect the nested Markov model through the lens of generalized probabilistic theory, an axiomatic framework to describe general physical theories. We prove that all the equality constraints defining the nested Markov model hold valid theory-independently. Yet, we show this model generally contains distributions not implementable even within such relaxed physical theories subjected to merely the relativity principles and mild probabilistic rules. To interpret the origin of such a gap, we establish a new causal model that defines valid distributions as projected from a high-dimensional Bell-type causal structure. The new model unveils inequality constraints induced by relativity principles, or equivalently high-dimensional conditional independences, which are absent in the nested Markov model. Nevertheless, we also notice that the restrictions on states and measurements introduced by the generalized probabilistic theory framework can pose additional inequality constraints beyond the new causal model. As a by-product, we discover a new causal structure exhibiting strict gaps between the distribution sets of a Bayesian network, generalized probabilistic theories, and the nested Markov model. We anticipate our results will enlighten further explorations on the unification of algebraic and physical perspectives of causality.

翻译：确定具有给定因果图的潜在概率分布对于因果性研究至关重要。为规避贝叶斯网络中隐变量表征的困难，嵌套马尔可夫模型通过精确列出观测变量的所有等式约束，提供了一种优雅的代数方法。然而，这种代数驱动的因果模型包含贝叶斯网络之外的分布，其物理解释仍不明确。本工作通过广义概率理论——一种描述一般物理理论的公理化框架——的视角审视嵌套马尔可夫模型。我们证明定义嵌套马尔可夫模型的所有等式约束在理论无关的意义上均成立。但我们同时表明，即使在仅遵循相对性原理及温和概率规则的松弛物理理论中，该模型通常也包含无法实现的分布。为解释这种差异的起源，我们建立了一个新的因果模型，将有效分布定义为高维贝尔型因果结构的投影。新模型揭示了由相对性原理（或等价的高维条件独立性）诱导的不等式约束，这些约束在嵌套马尔可夫模型中并不存在。然而，我们也注意到广义概率理论框架引入的状态与测量限制可能产生超出新因果模型的额外不等式约束。作为副产品，我们发现了一种新的因果结构，其展现出贝叶斯网络、广义概率理论与嵌套马尔可夫模型分布集合间的严格差异。我们预期这些结果将启发关于因果性代数视角与物理视角统一化的进一步探索。