Toward a Semiparametric Efficiency Theory under Equality Constraints in Nested Markov Models
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嵌套马尔可夫模型等式约束下的半参数效率理论探索
- 研究问题
- 潜变量有向无环图引出的 Verma 等式约束，如何转化为半参数模型的切空间限制，并用于刻画有效影响函数与效率界？
- 方法与思路
- 作者利用图固定操作后的加权条件矩表示，将约束转换为加权正交关系，并通过残差化矩函数及 Hilbert 空间投影刻画效率。
- 材料陈述的贡献
- 摘要称，文章刻画单个嵌套马尔可夫约束下的切空间正交补与效率界；多个约束仅得到正交补的一个子空间，完整刻画仍开放。
- 阅读关注点
- 核对加权矩表示与原约束的等价性、单约束切空间刻画的条件，以及多约束结果为何只覆盖子空间、对目标泛函有哪些要求。
- 阅读边界
- 这是 arXiv 预印本，同行评审及接收未核实；摘要未说明完整正则条件或可实施估计步骤，多约束的一般效率理论尚未完成。
原文摘要
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.