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Density Ratio-based Causal Discovery from Bivariate Continuous-Discrete Data

Takashi Nicholas Maeda, Shohei Shimizu, Hidetoshi Matsui

TL;DR

The paper tackles identifying causal direction between a continuous variable $X$ and a discrete variable $Y$ using observational data without strong distributional assumptions. It introduces a density-ratio based criterion, showing that the conditional density ratio $G_{c_s,c_t}(x)=\frac{p_{X|Y}(x|c_t)}{p_{X|Y}(x|c_s)}$ is monotonic when $X\to Y$, non-monotonic when $Y\to X$, and constant under no causation, enabling identifiability. The proposed DRCD method combines a KS test to verify causal existence, uLSIF-based density-ratio estimation, and a monotonicity test on the estimated ratio to infer direction, backed by theoretical identifiability results. Empirical results on synthetic and real-world data demonstrate DRCD's superior performance relative to existing mixed-data causal-discovery methods and its adherence to known domain constraints in practical datasets. This provides a principled, normalization-free approach for cross-type causal discovery in bivariate settings and lays groundwork for broader extensions.

Abstract

We propose a causal discovery method for mixed bivariate data consisting of one continuous and one discrete variable. Existing approaches either impose strong distributional assumptions or face challenges in fairly comparing causal directions between variables of different types, due to differences in their information content. We introduce a novel approach that determines causal direction by analyzing the monotonicity of the conditional density ratio of the continuous variable, conditioned on different values of the discrete variable. Our theoretical analysis shows that the conditional density ratio exhibits monotonicity when the continuous variable causes the discrete variable, but not in the reverse direction. This property provides a principled basis for comparing causal directions between variables of different types, free from strong distributional assumptions and bias arising from differences in their information content. We demonstrate its effectiveness through experiments on both synthetic and real-world datasets, showing superior accuracy compared to existing methods.

Density Ratio-based Causal Discovery from Bivariate Continuous-Discrete Data

TL;DR

The paper tackles identifying causal direction between a continuous variable and a discrete variable using observational data without strong distributional assumptions. It introduces a density-ratio based criterion, showing that the conditional density ratio is monotonic when , non-monotonic when , and constant under no causation, enabling identifiability. The proposed DRCD method combines a KS test to verify causal existence, uLSIF-based density-ratio estimation, and a monotonicity test on the estimated ratio to infer direction, backed by theoretical identifiability results. Empirical results on synthetic and real-world data demonstrate DRCD's superior performance relative to existing mixed-data causal-discovery methods and its adherence to known domain constraints in practical datasets. This provides a principled, normalization-free approach for cross-type causal discovery in bivariate settings and lays groundwork for broader extensions.

Abstract

We propose a causal discovery method for mixed bivariate data consisting of one continuous and one discrete variable. Existing approaches either impose strong distributional assumptions or face challenges in fairly comparing causal directions between variables of different types, due to differences in their information content. We introduce a novel approach that determines causal direction by analyzing the monotonicity of the conditional density ratio of the continuous variable, conditioned on different values of the discrete variable. Our theoretical analysis shows that the conditional density ratio exhibits monotonicity when the continuous variable causes the discrete variable, but not in the reverse direction. This property provides a principled basis for comparing causal directions between variables of different types, free from strong distributional assumptions and bias arising from differences in their information content. We demonstrate its effectiveness through experiments on both synthetic and real-world datasets, showing superior accuracy compared to existing methods.
Paper Structure (35 sections, 7 theorems, 87 equations, 2 figures, 2 tables)

This paper contains 35 sections, 7 theorems, 87 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Studies
  3. Model
  4. The model where $X$ is the cause of $Y$
  5. The model where $Y$ is the cause of $X$
  6. The model with no causal relationship
  7. Identifiability
  8. Method
  9. Experiments
  10. Experiments with Synthetic Data
  11. Experiments using real-world data
  12. Limitations
  13. Conclusion
  14. Proofs
  15. The proof of Lemma \ref{['lem:TSRV']}
  16. ...and 20 more sections

Key Result

Lemma 1

Let $W_1, W_2, \ldots, W_n$ be random variables that all satisfy the TSRV condition. If a random variable $W$ has probability density function where $\omega_i > 0$ and $\sum_{i=1}^{n} \omega_i = 1$, then $W$ also satisfies the TSRV condition.

Figures (2)

  • Figure 1: Monotonicity property of the density ratio. The figure illustrates our key finding: when a continuous variable ($X$) causes a discrete variable ($Y$) (Case 1, left), the conditional density ratio $\frac{P(X|Y=1)}{P(X|Y=0)}$ (red line) exhibits monotonic behavior. In contrast, when the causal direction is reversed and a discrete variable ($Y$) causes a continuous variable ($X$) (Case 2, right), this monotonicity property does not hold.
  • Figure 2: Inferred causal links between numerical (top) and categorical (bottom) variables in the UCI Heart Disease dataset. Methods were applied to all numerical-categorical pairs.

Theorems & Definitions (17)

  • Definition 1: Tail Symmetry of a Random Variable (TSRV)
  • Lemma 1: Closure of TSRV under mixtures
  • Definition 2: Monotonicity
  • Definition 3: Non-monotonicity
  • Lemma 2: Monotonicity of the density ratio under $X \to Y$
  • Lemma 3: Non-monotonicity of the density ratio under $Y \to X$
  • Lemma 4: Constant density ratio under no causal relationship
  • Theorem 1: Identifiability of the causal relationship
  • proof
  • proof
  • ...and 7 more