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Missing data and cluster graphs: cluster-level missingness vs variable-level missingness

NeurIPS 2026 · Probabilistic and scientific modeling › Causal discovery · approach: Identifiability with hidden confounders · 0 citations (OpenAlex)

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Introduces cluster-based missingness graphs (m-C-DMG and cm-C-DMG) and studies when joint distributions and macro causal effects remain recoverable under cluster-level abstractions of MNAR missingness.

Problem

Recoverability from missing data is usually analyzed with fully specified variable-level missingness graphs, yet in many applications only coarse cluster-level structural knowledge is available.

Approach

Extends m-graphs and cluster DMGs with two cluster-level missingness abstractions (variable-indexed m-C-DMG and cluster-merged cm-C-DMG), formalizes compatibility with variable-level m-ADMGs, shows the cm-C-DMG class contains the m-C-DMG class, and derives graphical conditions for joint recoverability and do-calculus-based macro effect recoverability.

Key results

• Proposition shows the compatibility class of an m-C-DMG is a strict subset of that of its associated cm-C-DMG, so cm-C-DMG recoverability implies m-C-DMG recoverability but not conversely.
• Theorem gives necessary and sufficient graphical condition for recovering the joint distribution over clusters: absence of a partially observed cluster adjacent to its R-vertex or connected to it via a collider-only path.
• Running HIV example: joint distribution and macro effect Pr(c_Y | do(c_X)) are shown recoverable via do-calculus rules and proxy substitution.

Surprising findings

• A macro causal effect can be recoverable even when the joint distribution is not recoverable.

Limitations

• Abstract graphs may be conservative; a query not recoverable from the abstraction may be recoverable in some compatible variable-level graph.
• cm-C-DMGs hide missingness distinctions that can be essential for recovery, so usefulness depends on the clustering.
• The cm-C-DMG considered only clusters missingness indicators when their variables share the same substantive cluster; the macro-effect conditions are sufficient only and completeness is open.

Experimental setup

Builds on: m-graphs C-DMG do-calculus ADMG d-separation

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