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Confidence in Causal Inference under Structure Uncertainty in Linear Causal Models with Equal Variances

David Strieder, Mathias Drton

TL;DR

It is argued that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal effects in order to give confidence regions for total causal effects that capture both sources of uncertainty: causal structure and numerical size of nonzero effects.

Abstract

Inferring the effect of interventions within complex systems is a fundamental problem of statistics. A widely studied approach employs structural causal models that postulate noisy functional relations among a set of interacting variables. The underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In a recent line of work, additional assumptions on the causal models have been shown to render this causal graph identifiable from observational data alone. One example is the assumption of linear causal relations with equal error variances that we will take up in this work. When the graph structure is known, classical methods may be used for calculating estimates and confidence intervals for causal effects. However, in many applications, expert knowledge that provides an a priori valid causal structure is not available. Lacking alternatives, a commonly used two-step approach first learns a graph and then treats the graph as known in inference. This, however, yields confidence intervals that are overly optimistic and fail to account for the data-driven model choice. We argue that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal effects. To address this issue, we present a framework based on test inversion that allows us to give confidence regions for total causal effects that capture both sources of uncertainty: causal structure and numerical size of nonzero effects.

Confidence in Causal Inference under Structure Uncertainty in Linear Causal Models with Equal Variances

TL;DR

It is argued that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal effects in order to give confidence regions for total causal effects that capture both sources of uncertainty: causal structure and numerical size of nonzero effects.

Abstract

Inferring the effect of interventions within complex systems is a fundamental problem of statistics. A widely studied approach employs structural causal models that postulate noisy functional relations among a set of interacting variables. The underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In a recent line of work, additional assumptions on the causal models have been shown to render this causal graph identifiable from observational data alone. One example is the assumption of linear causal relations with equal error variances that we will take up in this work. When the graph structure is known, classical methods may be used for calculating estimates and confidence intervals for causal effects. However, in many applications, expert knowledge that provides an a priori valid causal structure is not available. Lacking alternatives, a commonly used two-step approach first learns a graph and then treats the graph as known in inference. This, however, yields confidence intervals that are overly optimistic and fail to account for the data-driven model choice. We argue that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal effects. To address this issue, we present a framework based on test inversion that allows us to give confidence regions for total causal effects that capture both sources of uncertainty: causal structure and numerical size of nonzero effects.
Paper Structure (16 sections, 6 theorems, 43 equations, 6 figures, 1 table, 3 algorithms)

This paper contains 16 sections, 6 theorems, 43 equations, 6 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. Linear Structural Equation Models
  4. Causal Effects
  5. Confidence Intervals via Test Inversion
  6. Inversion of Tests
  7. Hypothesis of a Fixed Causal Effect
  8. Testing for Total Causal Effects
  9. Constrained Likelihood Ratio Test
  10. Split Likelihood Ratio Test
  11. Computational Details and Numerical Experiments
  12. Algorithm and Computational Shortcuts
  13. Maximum Likelihood Estimation
  14. Simulations
  15. Real Data Example
  16. ...and 1 more sections

Key Result

Proposition 3.2

For a covariance matrix $\Sigma \in$ PD$(d)$, the following statements are equivalent:

Figures (6)

  • Figure 1: Mean width of $95\%$-confidence intervals for the total causal effect of $X_1$ on $X_2$ in randomly generated $6$-dim. DAGs (1000 replications).
  • Figure 2: Percentages of times zero contained in $95\%$-confidence intervals in randomly generated $6$-dim. DAGs with non-zero effect (1000 replications). (Left) Against sample size. (Right) Against expected edge weights.
  • Figure 3: Mean width and percentages of times zero contained in $95\%$-confidence intervals for the total causal effect of $X_1$ on $X_2$ in randomly generated $12$-dim. DAGs (100 replications)
  • Figure 4: Mean computation times of $95\%$-confidence intervals for causal effect in randomly generated DAGs in seconds (100 replications). (Left) Against dimension. (Right) Against sample size.
  • Figure 5: Empirical coverage of $95\%$-confidence intervals for the total causal effect of $X_1$ on $X_2$ in randomly generated $6$-dim. DAGs under departure from equal error variances ($1000$ replications).
  • ...and 1 more figures

Theorems & Definitions (20)

  • Remark 2.1
  • Example 2.2
  • Remark 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • Remark 3.5
  • Example 3.6
  • ...and 10 more