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Causally Consistent Normalizing Flow

Qingyang Zhou, Kangjie Lu, Meng Xu

TL;DR

The paper tackles causal inconsistency between generative models and structured causal models, which can yield unfair outcomes. It introduces Causally Consistent Normalizing Flows (CCNF), leveraging a sequential SCM representation and partial causal transformations to preserve the SCM’s causal graph while maintaining multi-layer expressiveness, with at least $d$ layers where $d$ is the longest path length. CCNF is a multi-layer universal approximator and is causally consistent with the SCM, ensuring the induced graph $G_{GM}$ matches $G_M$ and enabling interventions and counterfactuals. Empirically, CCNF outperforms prior causally consistent GMs on synthetic benchmarks and improves real-world fairness on the German credit dataset, increasing accuracy from $73.00\%$ to $75.80\%$ and reducing individual unfairness from $9.00\%$ to $0.00\%$.

Abstract

Causal inconsistency arises when the underlying causal graphs captured by generative models like \textit{Normalizing Flows} (NFs) are inconsistent with those specified in causal models like \textit{Struct Causal Models} (SCMs). This inconsistency can cause unwanted issues including the unfairness problem. Prior works to achieve causal consistency inevitably compromise the expressiveness of their models by disallowing hidden layers. In this work, we introduce a new approach: \textbf{C}ausally \textbf{C}onsistent \textbf{N}ormalizing \textbf{F}low (CCNF). To the best of our knowledge, CCNF is the first causally consistent generative model that can approximate any distribution with multiple layers. CCNF relies on two novel constructs: a sequential representation of SCMs and partial causal transformations. These constructs allow CCNF to inherently maintain causal consistency without sacrificing expressiveness. CCNF can handle all forms of causal inference tasks, including interventions and counterfactuals. Through experiments, we show that CCNF outperforms current approaches in causal inference. We also empirically validate the practical utility of CCNF by applying it to real-world datasets and show how CCNF addresses challenges like unfairness effectively.

Causally Consistent Normalizing Flow

TL;DR

The paper tackles causal inconsistency between generative models and structured causal models, which can yield unfair outcomes. It introduces Causally Consistent Normalizing Flows (CCNF), leveraging a sequential SCM representation and partial causal transformations to preserve the SCM’s causal graph while maintaining multi-layer expressiveness, with at least layers where is the longest path length. CCNF is a multi-layer universal approximator and is causally consistent with the SCM, ensuring the induced graph matches and enabling interventions and counterfactuals. Empirically, CCNF outperforms prior causally consistent GMs on synthetic benchmarks and improves real-world fairness on the German credit dataset, increasing accuracy from to and reducing individual unfairness from to .

Abstract

Causal inconsistency arises when the underlying causal graphs captured by generative models like \textit{Normalizing Flows} (NFs) are inconsistent with those specified in causal models like \textit{Struct Causal Models} (SCMs). This inconsistency can cause unwanted issues including the unfairness problem. Prior works to achieve causal consistency inevitably compromise the expressiveness of their models by disallowing hidden layers. In this work, we introduce a new approach: \textbf{C}ausally \textbf{C}onsistent \textbf{N}ormalizing \textbf{F}low (CCNF). To the best of our knowledge, CCNF is the first causally consistent generative model that can approximate any distribution with multiple layers. CCNF relies on two novel constructs: a sequential representation of SCMs and partial causal transformations. These constructs allow CCNF to inherently maintain causal consistency without sacrificing expressiveness. CCNF can handle all forms of causal inference tasks, including interventions and counterfactuals. Through experiments, we show that CCNF outperforms current approaches in causal inference. We also empirically validate the practical utility of CCNF by applying it to real-world datasets and show how CCNF addresses challenges like unfairness effectively.

Paper Structure

This paper contains 27 sections, 12 theorems, 17 equations, 12 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A Motivating Example
  3. Preliminaries
  4. Structured Casual Model (SCM)
  5. Causal Normalizing Flows
  6. Causal Consistency
  7. Topological Batching
  8. Causally Consistent Normalizing Flows
  9. Definitions
  10. A Running Example
  11. CCNF: Properties and Operations
  12. Properties of CCNF
  13. Causal Inference Tasks
  14. Evaluation
  15. Causal Consistency
  16. ...and 12 more sections

Key Result

Theorem 5.1

Given a CCNF$T_{\theta}^\mathbf{B}$, for the $i$-th variable $X_i$, $X_i$ only depends on its parents $\mathbf{X}_{\mathbf{pa}_i}$ and $U_i$. Particularly assume $i \in \mathbf{B}_j$, we have $X_i = T_{\theta_j}^\mathbf{B_j}(U_i \mid \mathbf{X}_{\mathbf{pa}_i})$

Figures (12)

  • Figure 1: Prior distributions of causal NF and CCNF
  • Figure 2: The causal graph of an SCM describing an admission system with direct causalities that are intended (black solid line) and forbidden (red dashed line)
  • Figure 3: Causally consistent models and inconsistent models of prior works. G = Gender, A = age, S = Score, D = Decisions, M stands for nodes in the middle layer.
  • Figure 4: An example SCM, its topological order, and a related CCNF.
  • Figure 5: Causal graph for Non-linear Triangle dataset
  • ...and 7 more figures

Theorems & Definitions (16)

  • Theorem 5.1: Causality
  • Theorem 5.2: Universality
  • Theorem 5.3: Causal Consistency
  • Theorem 5.4: Minimum Layer
  • Lemma B.1
  • Theorem B.2
  • Lemma B.3: Composition
  • Lemma B.4: Partition
  • Theorem B.5: Causality
  • proof
  • ...and 6 more