有限状态嵌套马尔可夫模型中的递归头几何与无序高效推断
Recursive-Head Geometry and Order-Free Efficient Inference in Finite-State Nested Markov Models
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中文总结 AI 辅助
本文通过递归头几何证明有限状态嵌套马尔可夫模型在任意ADMG上的无序高效推断，提出规范梯度公式与有效估计量，并区分了无序有效性与多重稳健性。
中文摘要 AI 辅助
利用嵌套马尔可夫模型编码的等式约束需要其切空间几何。对于任意无环有向混合图（ADMGs）上的严格正有限状态模型，我们微分递归头图并证明其得分映射的值域是完整切空间。内在集合坐标块构成代数直和，不同区域（districts）的块是正交的，由此得到的Gram投影既不需要mb屏蔽性也不需要区域顺序。精确反例表明，仅凭观测中心化和核归一化并不能保证相切性。对于本文考虑的节点、完整源边和兼容路径特定干预目标，边界替换产生归一化配置有效律和无序规范梯度公式，并通过独立源重抽扩展到有限混合。一个具有精确余项恒等式的连贯一步估计量和模型有效的图流目标最大似然估计量在所述局部条件下是有效的。在一个更窄的源隔离固定节点子类上，顺序估计量具有精确转移因子化漂移，并在m个活动区域转移上具有多达$2^m$个干扰校正机制；其全正确影响函数投影到规范梯度上，且精确有理律表现出严格方差差距。这些结果将任意有限状态ADMG上的无序有效性与更窄构造的多重稳健性区分开来。
英文摘要
Exploiting the equality restrictions that nested Markov models encode requires their tangent-space geometry. For strictly positive finite-state models on arbitrary acyclic directed mixed graphs (ADMGs), we differentiate the recursive-head chart and prove that the range of its score map is the full tangent space. Intrinsic-set coordinate blocks form an algebraic direct sum, blocks of distinct districts are orthogonal, and the resulting Gram projection needs neither mb-shieldedness nor a district order. Exact counterexamples show that observational centering and kernel normalization alone do not certify tangency. For the node, complete-source edge, and compatible path-specific intervention targets considered here, boundary substitution yields a normalized configured active law and an order-free canonical-gradient formula, extended to finite mixtures by independent source redraws. A coherent one-step estimator with an exact remainder identity and a model-valid chart-flow targeted maximum likelihood estimator are efficient under stated local conditions. On a narrower source-isolated fixed-node subclass, a sequential estimator has an exact transition-factorized drift and up to $2^m$ nuisance-correctness regimes over $m$ active-district transitions; its all-correct influence function projects onto the canonical gradient, and an exact rational law exhibits a strict variance gap. These results separate order-free efficiency on arbitrary finite-state ADMGs from multiple robustness of a narrower construction.
