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Omitted Labels Induce Nontransitive Paradoxes in Causality

Bijan Mazaheri, Siddharth Jain, Matthew Cook, Jehoshua Bruck

TL;DR

This work analyzes how omitting label contexts in data can produce irrecoverable causal effects and lead to nontransitive, paradoxical conclusions. By connecting omitted-label causality to the Condorcet paradox, it formalizes a correspondence between networks of contexts and aggregations of rankings (AR) and soft rankings (ASR) within the linear ordering polytope. The core result shows ARs and ASRs occupy the same convex edge-weight space, enabling a unified view of consistency across partial studies and decision fusion across contexts. The framework offers bounds and diagnostics for inconsistencies when fusing conclusions from different label contexts, with implications for rare-disease studies and model-based decision fusion.

Abstract

We explore "omitted label contexts," in which training data is limited to a subset of the possible labels. This setting is standard among specialized human experts or specific, focused studies. By studying Simpson's paradox, we observe that ``correct'' adjustments sometimes require non-exchangeable treatment and control groups. A generalization of Simpson's paradox leads us to study networks of conclusions drawn from different contexts, within which a paradox of nontransitivity arises. We prove that the space of possible nontransitive structures in these networks exactly corresponds to structures that form from aggregating ranked-choice votes.

Omitted Labels Induce Nontransitive Paradoxes in Causality

TL;DR

This work analyzes how omitting label contexts in data can produce irrecoverable causal effects and lead to nontransitive, paradoxical conclusions. By connecting omitted-label causality to the Condorcet paradox, it formalizes a correspondence between networks of contexts and aggregations of rankings (AR) and soft rankings (ASR) within the linear ordering polytope. The core result shows ARs and ASRs occupy the same convex edge-weight space, enabling a unified view of consistency across partial studies and decision fusion across contexts. The framework offers bounds and diagnostics for inconsistencies when fusing conclusions from different label contexts, with implications for rare-disease studies and model-based decision fusion.

Abstract

We explore "omitted label contexts," in which training data is limited to a subset of the possible labels. This setting is standard among specialized human experts or specific, focused studies. By studying Simpson's paradox, we observe that ``correct'' adjustments sometimes require non-exchangeable treatment and control groups. A generalization of Simpson's paradox leads us to study networks of conclusions drawn from different contexts, within which a paradox of nontransitivity arises. We prove that the space of possible nontransitive structures in these networks exactly corresponds to structures that form from aggregating ranked-choice votes.
Paper Structure (16 sections, 4 theorems, 11 equations, 3 figures, 3 tables)

This paper contains 16 sections, 4 theorems, 11 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Domain Adaptation
  3. Omitted Label Contexts
  4. Summary
  5. Contributions
  6. Outline
  7. Notation
  8. Simpson's Paradox
  9. Omitted Label Contexts
  10. Causality within Omitted Label Contexts
  11. Networks of Contexts
  12. The Condorcet Paradox
  13. Cardinal vs Ordinal Systems
  14. Aggregations of Rankings and Soft Rankings
  15. Probabilities can Emulate Preferences
  16. ...and 1 more sections

Key Result

Theorem 7

$\text{Co}(\mathcal{A})$ and $\overline{\text{Co}}(\mathcal{B})$ are the same.

Figures (3)

  • Figure 1: (a) A causal DAG depicting confounding from a common cause $X$. (b) The causal DAG that "severs" $X \rightarrow T$ by reweighting for exchangeability. (c) The causal DAG depicting the effect of an omitted label context $C$, which has been conditioned on.
  • Figure 2: The Condorcet paradox as an aggregation of rankings.
  • Figure 3:

Theorems & Definitions (10)

  • Definition 1: Ranking
  • Definition 2: Aggregation of Rankings (AR)
  • Definition 3: Aggregate Preference
  • Definition 4: Soft Rankings
  • Definition 5: Aggregation of Soft Rankings (ASR)
  • Definition 6: Aggregate Probability
  • Theorem 7
  • Lemma 8
  • Lemma 9
  • Lemma 10