Causal Foundation Models with Continuous Treatments

Causal inference is the practice of estimating the effect of a treatment or intervention from data. It traditionally requires a bespoke pipeline for every new problem: first proposing a causal mechanism, selecting a compatible estimator, and finally training it. Meanwhile, across diverse settings and modalities, much of machine learning has shifted to the paradigm of foundation models: networks pretrained once at scale and applied to new tasks without fine-tuning. Causal foundation models (CFMs) bring this paradigm to causal inference. CFMs are pretrained neural networks that estimate causal quantities, such as the average treatment effect, on entirely new datasets using in-context learning without requiring model updates. This work provides a practical introduction to this emerging area. We summarize the necessary background in causal inference and machine learning before discussing CFMs. Throughout, we include example code and Jupyter notebooks.

Extending discrete-time causal Prior-data Fitted Networks for time series to continuous time invites writing the mechanism as a stochastic differential equation (SDE) -- but if the SDE is integrated \emph{once per observation gap}, the trajectory law depends on when it is observed, and the prior remains a discrete-time Markov model in SDE clothing. We propose a precise continuity criterion -- trajectory-law invariance to the observation schedule -- together with a three-tier taxonomy (discrete; naive observation-grid integration; fine-grid integration with decoupled observation) and a construction realising the top tier on a random DAG with OU or small-MLP nonlinear drifts, irregular observation schedules, and hard / soft / time-varying interventions. A $2 \times 2$ encoder $\times$ integrator ablation, run independently on a linear and a nonlinear prior, finds fine-grid integration beats naive on 8/8 cells (sign-consistency $p < 1/256$) with the gap growing as the eval grid refines; the encoder axis is null with fine integration but time-aware-leading with naive. We release the prior and a preliminary zero-shot protocol on pharmacokinetic and physical-system data.

Causal effect estimation from observational data requires careful adjustment for confounding. Classical estimators such as inverse probability weighting and augmented inverse probability weighting are effective under favorable model specification, but may become unstable when treatment assignment and outcome mechanisms are complex, non-linear, and high-dimensional. Machine learning and representation learning approaches improve flexibility, yet joint training can allow outcome-related information to influence treatment-side representations, which is undesirable from a causal perspective. We propose MOCA (Modular One-way Causal Attention), a transformer-based framework that separates treatment and outcome modeling through a modular design, and performs confounder adjustment using a one-way attention mechanism. A cutting-feedback strategy, implemented via gradient detachment, prevents the outcome loss from updating the treatment module. This design preserves directional information flow while retaining the representational power of transformer architectures for causal inference. Across multiple simulated scenarios, including linear, nonlinear, heavy-tailed, hidden confounding, and high-dimensional settings, MOCA shows competitive or improved performance relative to IPW, AIPW, X-learner, TARNet, and DragonNet. We further illustrate the method on the Infant Health and Development Program dataset and the Dehejia-Wahba dataset as real-world benchmarks. These results suggest that modular attention with one-way information flow provides a promising and interpretable direction for causal inference with modern deep learning models.