Long-term Causal Inference via Modeling Sequential Latent Confounding

Abstract

Long-term causal inference is an important but challenging problem across various scientific domains. To solve the latent confounding problem in long-term observational studies, existing methods leverage short-term experimental data. Ghassami et al. propose an approach based on the Conditional Additive Equi-Confounding Bias (CAECB) assumption, which asserts that the confounding bias in the short-term outcome is equal to that in the long-term outcome, so that the long-term confounding bias and the causal effects can be identified. While effective in certain cases, this assumption is limited to scenarios where there is only one short-term outcome with the same scale as the long-term outcome. In this paper, we introduce a novel assumption that extends the CAECB assumption to accommodate temporal short-term outcomes. Our proposed assumption states a functional relationship between sequential confounding biases across temporal short-term outcomes, under which we theoretically establish the identification of long-term causal effects. Based on the identification result, we develop an estimator and conduct a theoretical analysis of its asymptotic properties. Extensive experiments validate our theoretical results and demonstrate the effectiveness of the proposed method.

Figures (5)

• Figure 1: Causal graphs for experimental and observational data with $\mathbf{X}$ as covariates, $\mathbf{U}$ as latent confounders, $A$ as treatment, $\mathbf{S}$ as short-term outcomes, and $Y$ as the long-term outcome. Fig. \ref{['fig: causal graph']}(a) shows the experimental data, where $A$ is unaffected by $\mathbf{U}$ and $Y$ cannot be unobserved. Fig. \ref{['fig: causal graph']}(b) depicts the observational data, where $\mathbf{U}$ affects $A$, $\mathbf{S}$, and $Y$, and $Y$ can be observed. Fig. \ref{['fig: causal graph']}(c) illustrates the full causal graph with temporally extended short-term outcomes.
• Figure 2: Schematic representations of CAECB assumption proposed by ghassami2022combining and our FCAECB assumption. As shown in Figure \ref{['fig: existing assumption']}, the CAECB assumption requires that the confounding bias in the short-term outcome is equal to that in the long-term outcome. As shown in Figure \ref{['fig: our assumption']}, the FCAECB assumption relaxes this constraint by allowing for temporal short-term outcomes and only requiring that confounding biases across different time steps follow a specific pattern rather than remaining equal.
• Figure 3: Results of the experiments in terms of different choice of $\mu$ and $T$.
• Figure 4: Results of control experiments with fixed $T$ and varying $\mu$ (Figures \ref{['fig: PEHE varyT']} and \ref{['fig: MAE varyT']}), and with fixed $\mu$ and varying $T$ (Figures \ref{['fig: PEHE varymu']} and \ref{['fig: MAE varymu']}).
• Figure 5: Results of control experiments with varying sample sizes $n_e$ and $n_o$.

Theorems & Definitions (24)

• Theorem 3.7
• Remark 3.8: Interpretation on Assumption \ref{['assum: equ bias']}
• Remark 4.2: Interpretation on Assumption \ref{['assum: time series equ bias']}
• Proposition 4.3
• Remark 4.4: Insight of Assumption \ref{['theo: time series identifi']}
• Remark 4.5: Testing Assumption \ref{['theo: time series identifi']} over Short Term Duration
• Example 4.6
• Theorem 4.7
• Definition 5.1: Hölder ball
• Lemma 5.4