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

2026-07-25

AI Summary

This work addresses a fundamental limitation in existing semiparametric inference methods for non-DAG Markov models, where the orthogonal complement of the tangent space lacks an explicit characterization. By integrating semiparametric statistical theory, influence function analysis, and the conditional independence structure inherent in graphical models, the paper establishes, for the first time, a general closed-form expression for the orthogonal complement of the tangent space in broad classes of Markov models—including undirected graphs, chain graphs, and directed acyclic mixed graphs. Building on this characterization, the authors fully describe the class of influence functions for target parameters such as conditional means, thereby enabling efficient semiparametric estimators that are root-n consistent and asymptotically normal.

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 finite-dimensional target parameters 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.

Problem

• Markov models
• tangent space
• orthogonal complement
• semiparametric inference
• graphical models

Innovation

• semiparametric inference
• influence functions
• tangent space
• Markov models
• graphical models