Deriving Complete Constraints in Hidden Variable Models

TL;DR

This work addresses deriving the complete set of observable constraints implied by hidden-variable causal models with discrete observed variables. It develops a systematic method that translates the latent structure into response-function variables and uses linear functional constraints, subsequently converting to the observable probability space via vertex-to-halfspace representations. The authors prove completeness for graphs whose districts all have $c$-degree $1$, and illustrate the approach with novel examples including sequential instrumental variables, nested equivalent models with distinct inequalities, and tripartite Bell graphs; they also discuss extensions to incomplete linear constraints and observationally equivalent DAGs, and provide public software. The results enable falsification and tighter inference under hidden-variable assumptions, with potential applications in causal discovery and constrained estimation. Overall, the paper generalizes complete observable-constraint derivation beyond classical IV, introducing MERA-type constraints that can include both inequalities and equalities and offering a practical computational framework.”

Abstract

Hidden variable graphical models can sometimes imply constraints on the observable distribution that are more complex than simple conditional independence relations. These observable constraints can falsify assumptions of the model that would otherwise be untestable due to the unobserved variables and can be used to constrain estimation procedures to improve statistical efficiency. Knowing the complete set of observable constraints is thus ideal, but this can be difficult to determine in many settings. In models with categorical observed variables and a joint distribution that is completely characterized by linear relations to the unobservable response function variables, we develop a systematic method for deriving the complete set of observable constraints. We illustrate the method in several new settings, including ones that imply both inequality and equality constraints.

Figures (12)

• Figure 1: Example to illustrate Definitions \ref{['def:districts']} and \ref{['def:cdeg']}. The left figure is the original one, and the right figure has the edges originating from observed variables removed. The graph has three districts: $\{G\}$ and $\{A, C, E\}$ with c-degrees 1 and $\{B, D, F\}$ with c-degree 2.
• Figure 2: A sequential instrumental variable example, and its response function variable transformation.
• Figure 3: The instrumental variable DAG in original for (left), showing the two districts (center), and response variable form (right).
• Figure 4: Front door model.
• Figure 5: Two graphs that are nested Markov equivalent, but which have distinct inequality constraints.

Theorems & Definitions (13)

• Definition 1
• Definition 2
• Definition 3
• Remark 1
• Proposition 1
• Proposition 2
• Proposition 3
• Definition 4
• Proposition 4
• Remark 2