Recursive-Head Geometry and Order-Free Efficient Inference in Finite-State Nested Markov Models
Abstract
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 2m 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.
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BibTeX
@misc{wei2026recursivehead,
title = {{Recursive-Head Geometry and Order-Free Efficient Inference in Finite-State Nested Markov Models}},
author = {Haoyu Wei},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17939},
eprint = {2609.17939},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17939},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Recursive-Head Geometry and Order-Free Efficient Inference in Finite-State Nested Markov Models
AU - Haoyu Wei
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17939
UR - https://arxiv.org/abs/2609.17939
AB - 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.
ER -