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Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions

Naoya Hashimoto, Yuta Kawakami, Jin Tian

Abstract

Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.

Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions

Abstract

Evaluating joint probabilities of potential outcomes and observed variables, and their linear combinations, is a fundamental challenge in causal inference. This paper addresses the bounding and identification of these probabilities in settings with discrete treatment and discrete ordinal outcome. We propose new families of monotonicity assumptions and formulate the bounding problem as a linear programming problem. We further introduce a new monotonicity assumption specifically to achieve identification. Finally, we present numerical experiments to validate our methods and demonstrate their application using real-world datasets.
Paper Structure (27 sections, 14 theorems, 124 equations, 3 figures, 25 tables)

This paper contains 27 sections, 14 theorems, 124 equations, 3 figures, 25 tables.

Table of Contents

  1. INTRODUCTION
  2. MOTIVATING EXAMPLE
  3. NOTATIONS AND BACKGROUND
  4. MONOTONICITY ASSUMPTIONS FOR DISCRETE TREATMENT AND OUTCOME
  5. BOUNDING WITH MONOTONICITY ASSUMPTIONS
  6. Objective of the Study
  7. Bounding Problem Setup
  8. Bounding Joint POs/OVs Probabilities and Their Linear Combinations From Finite Samples
  9. IDENTIFICATION WITH MONOTONICITY ASSUMPTIONS
  10. Identifying Joint POs/OVs Probabilities and Their Linear Combinations
  11. Estimating Joint POs/OVs Probabilities and Their Linear Combinations
  12. APPLICATION TO REAL-WORLD DATASETs
  13. Experimental Data
  14. Observational Data
  15. CONCLUSION
  16. ...and 12 more sections

Key Result

Proposition 1

We have the following relationships among the MAs presented thus far: (1) Assumptions monobin_s through monomatASS are all special cases of Assumption assum_mono_with_prob_s. (2) Assumption mono_it is a special case of Assumptions weak_mono-bi, monomatAS, and monomatASS. (3) Assumption monobin_w is

Figures (3)

  • Figure 1: Results of the estimates of the bounds of $\mathbb{P}(Y_0=0,Y_1=0,Y_2=1)$ as $L$ varies. The x-axis shows the value of $L$, and the y-axis shows the mean estimates of the bounds of $\mathbb{P}(Y_0=0,Y_1=0,Y_2=1)$.
  • Figure 2: Results of the estimates of the bounds of $\mathbb{E}[Y_1-Y_0|X=2,Y=2]$ as $L$ varies. The x-axis shows the value of $L$, and the y-axis shows the mean estimates of the bounds of $\mathbb{E}[Y_1-Y_0|X=2,Y=2]$.
  • Figure 3: Results of the estimates of the bounds of $\mathbb{E}[(Y_1-Y_0)^2]$ as $L$ varies. The x-axis shows the value of $L$, and the y-axis shows the mean estimates of the bounds of $\mathbb{E}[(Y_1-Y_0)^2]$.

Theorems & Definitions (28)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Theorem 2: Identifiability under MA
  • Theorem 3: Identifiability under MA
  • proof
  • proof
  • proof
  • proof
  • Lemma 1
  • ...and 18 more