AI 中文总结
该研究针对嵌套马尔可夫模型的Verma约束，建立半参数效率理论框架，通过图形固定操作的加权正交关系刻画切空间，为嵌套马尔可夫模型的效率分析提供了几何基础。
AI 中文摘要
带有潜变量的有向无环图（DAG）的概率模型，在普通条件独立性之外，对观测数据分布施加了等式约束。这些被称为Verma约束，出现在与无向混合图（ADMG）相关的嵌套马尔可夫模型中，而ADMG是潜变量DAG的潜投影。尽管嵌套马尔可夫模型已从图形表示和因果识别角度得到广泛研究，但它们对半参数效率理论的影响仍鲜为人知。我们开发了用于建立由Verma约束定义的统计模型的半参数框架的相关结果。我们的关键发现是，嵌套马尔可夫约束在由图形固定操作诱导的后固定分布下，允许加权条件矩表示。我们证明，固定操作在L2(P)中诱导加权正交关系，从而将Verma约束转化为显式切空间限制。基于此表示，我们通过残差化加权矩函数，刻画了由单个嵌套马尔可夫约束定义的模型的切空间正交补。这种几何表述通过正交投影和等效最小方差公式，得到了半参数有效影响函数和效率界的希尔伯特空间刻画。我们进一步讨论了对涉及多个嵌套马尔可夫约束的模型的扩展，对于此类模型，我们将正交补的一个子空间刻画为相应加权正交关系的和，而完整的切空间刻画仍待解决。更广泛地说，我们的结果将嵌套图形结构与半参数希尔伯特空间几何联系起来，为嵌套马尔可夫模型的一般效率理论提供了基础。我们通过多个潜变量DAG来说明该框架。
英文摘要
Probabilistic models of Directed Acyclic Graphs (DAGs) with latent variables impose equality constraints on the observed data distribution beyond ordinary conditional independencies. These so-called Verma constraints arise in nested Markov models associated with Acyclic Directed Mixed Graphs, the latent projection of latent-variable DAGs. While nested Markov models have been extensively studied from the perspectives of graphical representation and causal identification, their implications for semiparametric efficiency theory remain less understood. We develop results toward establishing a semiparametric framework for statistical models defined by Verma constraints. Our key observation is that nested Markov constraints admit weighted conditional-moment representations under post-fixing distributions induced by graphical fixing operations. We show that fixing induces weighted orthogonality relations in L2(P), thereby converting Verma constraints into explicit tangent-space restrictions. Building on this representation, we characterize the tangent-space orthocomplement for models defined by a single nested Markov constraint through residualized weighted moment functions. This geometric formulation yields Hilbert-space characterizations of semiparametric efficient influence functions and efficiency bounds via orthogonal projection and equivalent minimum-variance formulations. We further discuss extensions to models involving multiple nested Markov constraints, for which we characterize a subspace of the orthocomplement as sums of the corresponding weighted orthogonality relations, while leaving the complete tangent-space characterization open. More broadly, our results connect nested graphical structure with semiparametric Hilbert-space geometry and provide a foundation for a general efficiency theory for nested Markov models. We illustrate the framework through several latent-variable DAGs.
