Bayesian Intervention Optimization for Causal Discovery

TL;DR

This work tackles efficient causal discovery by integrating active interventions with Bayes-factor–based hypothesis testing. It defines a decision-theoretic objective, $P_{DC}$, that quantifies the probability of obtaining decisive and correct evidence after an intervention and uses Bayesian optimization to select the most informative $\mathrm{do}(X=x)$. The methodology combines observational priors, do-calculus, Bayes factors, and Monte Carlo–based estimation to update hypotheses and refine interventions iteratively. Empirical results on synthetic data show superior performance to information-gain baselines for challenging causal structures, highlighting the practical value of targeted, Bayes-factor–driven active interventions in causal discovery.

Abstract

Causal discovery is crucial for understanding complex systems and informing decisions. While observational data can uncover causal relationships under certain assumptions, it often falls short, making active interventions necessary. Current methods, such as Bayesian and graph-theoretical approaches, do not prioritize decision-making and often rely on ideal conditions or information gain, which is not directly related to hypothesis testing. We propose a novel Bayesian optimization-based method inspired by Bayes factors that aims to maximize the probability of obtaining decisive and correct evidence. Our approach uses observational data to estimate causal models under different hypotheses, evaluates potential interventions pre-experimentally, and iteratively updates priors to refine interventions. We demonstrate the effectiveness of our method through various experiments. Our contributions provide a robust framework for efficient causal discovery through active interventions, enhancing the practical application of theoretical advancements.

Figures (6)

• Figure 1: Results under different ground truths with $k_0 = \frac{1}{k_1} = 10$: $P_{DC}$, $\log \text{BF}_{01}$, and $P(\mathbb{H}_{gt} \mid \mathbf{D}_\text{int})$. The first row corresponds to $\mathbb{H}_0$ ($X \gets Y$)). The second row corresponds to $\mathbb{H}_0$ ($X \gets U\to Y$), and the last row corresponds to $\mathbb{H}_1$ ($X \to Y$).
• Figure 2: Results under different ground truths with $k_0 = \frac{1}{k_1} = 10$: $P_{DC}$, $\log \text{BF}_{01}$, and $P(\mathbb{H}_{gt} \mid \mathbf{D}_\text{int})$. But with random mean and random covariance compared to the settings in the Figure \ref{['fig:results_h0_h1']}.The first row corresponds to $\mathbb{H}_0$ ($X \gets Y$)). The second row corresponds to $\mathbb{H}_0$ ($X \gets U\to Y$), and the last row corresponds to $\mathbb{H}_1$ ($X \to Y$).
• Figure 3: Results under different ground truths with $k_0 = \frac{1}{k_1} = 10$: $P_{DC}$, $\log \text{BF}_{01}$, and $P(\mathbb{H}_{gt} \mid \mathbf{D}_\text{int})$. The first row corresponds to $\mathbb{H}_0$ ($X \gets Y$)). The second row corresponds to $\mathbb{H}_0$ ($X \gets U\to Y$), and the last row corresponds to $\mathbb{H}_1$ ($X \to Y$).
• Figure 4: Results under different ground truths with $k_0 = \frac{1}{k_1} = 10$: $P_{DC}$, $\log \text{BF}_{01}$, and $P(\mathbb{H}_{gt} \mid \mathbf{D}_\text{int})$. But with random mean and random covariance compared to the settings in the Figure \ref{['fig:k030results_h0_h1']}. The first row corresponds to $\mathbb{H}_0$ ($X \gets Y$)). The second row corresponds to $\mathbb{H}_0$ ($X \gets U\to Y$), and the last row corresponds to $\mathbb{H}_1$ ($X \to Y$).
• Figure 5: Results under different ground truths with $k_0 = \frac{1}{k_1} = 10$: $P_{DC}$, $\log \text{BF}_{01}$, and $P(\mathbb{H}_{gt} \mid \mathbf{D}_\text{int})$. The first row corresponds to $\mathbb{H}_0$ ($X \gets Y$)). The second row corresponds to $\mathbb{H}_0$ ($X \gets U\to Y$), and the last row corresponds to $\mathbb{H}_1$ ($X \to Y$).