Stable Blanket with Hidden Variables and Cycles

TL;DR

This paper extends the graphical theory of stable predictors to complex causal models with hidden variables and cycles, providing new characterizations and conditions for invariance across environments.

Contribution

It introduces novel graphical characterizations of stable blankets in models with hidden variables and cycles, combining ADMGs, DGs, and $\sigma$-separation.

Findings

Graphical criteria for Markov blankets in models with hidden variables.

Conditions for stable predictor sets under interventions.

Extension of stabilized regression theory to cyclic and latent-variable models.

Abstract

Stabilized regression aims to identify a set of predictors whose conditional relationship with a response variable remains invariant across different environments. Existing graphical characterizations of the stable blanket are mainly developed for structural causal models (SCMs) without hidden variables or causal cycles. However, latent variables and feedback relationships naturally arise in many applications, and they can change both the Markov blanket and the set of predictors that remain stable under interventions. This paper studies stable blankets in graphical causal models with hidden variables, causal cycles, and both features simultaneously. For models with hidden variables, we use acyclic directed mixed graphs (ADMGs) and $m$-separation to characterize the Markov blanket and to construct intervention-stable predictor sets. We introduce the notion of an intervened sub-district and use it to describe how interventions may affect districts connected to the response. For models with cycles, we work with directed graphs (DGs) and directed mixed graphs (DMGs) together with $\sigma$-separation, treating strongly connected components (SCCs) as the basic graphical units. We then combine these ideas to analyze models with both hidden variables and cycles. The main results give graphical characterizations of Markov blankets, stable frontiers, and stable blankets in these generalized settings. In particular, we identify conditions under which the response is conditionally independent of intervention variables given a suitable predictor set, and we describe when such sets are minimal or unique. These results extend the graphical interpretation of stabilized regression beyond acyclic fully observed models.