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Causal and Counterfactual Views of Missing Data Models

Razieh Nabi, Rohit Bhattacharya, Ilya Shpitser, James M. Robins

TL;DR

It is made explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had the authors (possibly contrary to fact) been able to observe them.

Abstract

It is often said that the fundamental problem of causal inference is a missing data problem -- the comparison of responses to two hypothetical treatment assignments is made difficult because for every experimental unit only one potential response is observed. In this paper, we consider the implications of the converse view: that missing data problems are a form of causal inference. We make explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had we (possibly contrary to fact) been able to observe them. Drawing analogies with causal inference, we show how identification assumptions in missing data can be encoded in terms of graphical models defined over counterfactual and observed variables. We review recent results in missing data identification from this viewpoint. In doing so, we note interesting similarities and differences between missing data and causal identification theories.

Causal and Counterfactual Views of Missing Data Models

TL;DR

It is made explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had the authors (possibly contrary to fact) been able to observe them.

Abstract

It is often said that the fundamental problem of causal inference is a missing data problem -- the comparison of responses to two hypothetical treatment assignments is made difficult because for every experimental unit only one potential response is observed. In this paper, we consider the implications of the converse view: that missing data problems are a form of causal inference. We make explicit how the missing data problem of recovering the complete data law from the observed law can be viewed as identification of a joint distribution over counterfactual variables corresponding to values had we (possibly contrary to fact) been able to observe them. Drawing analogies with causal inference, we show how identification assumptions in missing data can be encoded in terms of graphical models defined over counterfactual and observed variables. We review recent results in missing data identification from this viewpoint. In doing so, we note interesting similarities and differences between missing data and causal identification theories.
Paper Structure (19 sections, 2 theorems, 34 equations, 9 figures)

This paper contains 19 sections, 2 theorems, 34 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Missing Data Models
  3. Classical Missing Data Models
  4. Counterfactual Views of Classical Missing Data Models
  5. Identification in Missing Data Models
  6. Directed Acyclic Graphs in Causal Inference
  7. Statistical DAG Models
  8. Causal DAG Models
  9. Missing Data DAG Models
  10. Hierarchy of missing data DAG models
  11. Examples of missing data DAG models
  12. Identification in Missing Data DAG Models
  13. Sequential Interventions
  14. Parallel Interventions
  15. Sequential and Parallel Interventions
  16. ...and 4 more sections

Key Result

Proposition 1

Under the missingness model of an m-DAG $\mathcal{G}_m$

Figures (9)

  • Figure 1: (a) A DAG where $U_1$ and $U_2$ may be unmeasured; (b) A conditional DAG illustrating interventions on $R_1$ and $R_2$.
  • Figure 2: Examples of missing data DAG models.
  • Figure 3: An illustration of the operation of a sequential identification algorithm. (a) Permutation model; (b) Intervention on $R_2$; (c) Intervention on $R_1$ after $R_2$.
  • Figure 4: An illustration of the operation of a parallel identification algorithm. (a) Block-parallel; (b) Intervention on $R_2$ and selection on $R_1$; (c) Intervention on $R_1$ and selection on $R_2$.
  • Figure 5: (a) An example m-DAG corresponding to a model where interventions must be applied both sequentially and in parallel to yield identification; (b) Graph derived from (a) representing the intermediate step of the identification algorithm where $R_2$ and $R_3$ are simultaneously set to $1$.
  • ...and 4 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Lemma 1: Invariance property