A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Models

Trung Q. Phung, Ilya Shpitser
Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, PMLR 337:5448-5470, 2026.

Abstract

Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of {Markov} models, defined using solely conditional independence (CI) restrictions. In order to estimate a finite-dimensional target parameter in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are {Markov} relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for {Markov} models not equivalent to a {DAG} model—such as ordinary {Markov} models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs—the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general {Markov} models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.

Cite this Paper

BibTeX
@InProceedings{pmlr-v337-phung26a, title = {A Characterization of the Orthocomplement of the Tangent Space of Semiparametric {Markov} Models}, author = {Phung, Trung Q. and Shpitser, Ilya}, booktitle = {Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence}, pages = {5448--5470}, year = {2026}, editor = {Perković, Emilija and Malinsky, Daniel}, volume = {337}, series = {Proceedings of Machine Learning Research}, month = {17--21 Aug}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v337/main/assets/phung26a/phung26a.pdf}, url = {https://proceedings.mlr.press/v337/phung26a.html}, abstract = {Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of {Markov} models, defined using solely conditional independence (CI) restrictions. In order to estimate a finite-dimensional target parameter in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are {Markov} relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for {Markov} models not equivalent to a {DAG} model—such as ordinary {Markov} models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs—the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general {Markov} models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.} }
Endnote
%0 Conference Paper %T A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Models %A Trung Q. Phung %A Ilya Shpitser %B Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2026 %E Emilija Perković %E Daniel Malinsky %F pmlr-v337-phung26a %I PMLR %P 5448--5470 %U https://proceedings.mlr.press/v337/phung26a.html %V 337 %X Graphical models are ubiquitous in social and empirical science as they are intuitive and easy to use. These models belong to the broader class of {Markov} models, defined using solely conditional independence (CI) restrictions. In order to estimate a finite-dimensional target parameter in such models efficiently, semi-parametric theory provides a principled framework for constructing regular and asymptotically linear estimators via influence functions (IFs). These estimators are asymptotically normal and root-$n$ consistent. Characterizing the class of all influence functions for a target parameter is crucial for statistically efficient inference in these models. For models that are {Markov} relative to directed acyclic graphs (DAGs), the orthogonal complement of the tangent space is known, implying that for any target the class of all influence functions can be derived once an influence function is obtained. On the other hand, for {Markov} models not equivalent to a {DAG} model—such as ordinary {Markov} models associated with undirected graphs, chain graphs, or acyclic directed mixed graphs—the orthogonal complement has not been characterized, impeding semi-parametric inference in these models. We derive closed form expressions for the orthogonal complement of the tangent space for general {Markov} models and illustrate our results by characterizing the class of influence functions for the conditional mean parameter in several graphical models.
APA
Phung, T.Q. & Shpitser, I.. (2026). A Characterization of the Orthocomplement of the Tangent Space of Semiparametric Markov Models. Proceedings of the 42nd Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research 337:5448-5470 Available from https://proceedings.mlr.press/v337/phung26a.html.

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