ProximalFM: Amortized Proximal Causal Inference under Hidden Confounding

Standard causal identification methods often assume no unmeasured confounding and can fail when relevant confounders are unobserved. Proximal causal inference instead uses proxy variables to identify effects under hidden confounding and additional assumptions. However, nonparametric proximal estimation can be challenging in practice: recovering causal estimands such as the conditional average treatment effect (CATE) requires solving an ill-posed integral equation that is data-hungry, hyperparameter-sensitive, and optimization-unstable. Bayesian inference for such models provides a desirable alternative, mitigating these difficulties by regularizing through the prior. However, computing a posterior is itself challenging, as a typical likelihood function will include latent variables. Following the recent success of tabular foundation models in backdoor, instrumental variable, and frontdoor settings, we propose that prior-data fitted networks (PFNs) are uniquely suited to resolve this bottleneck. Indeed, by training on synthetic data sampled from compliant structural causal models with access to oracle counterfactuals, we simplify the task substantially, amortizing the implied Bayesian operator inversion into a single transformer forward pass. Compared to prior literature that focuses primarily on point estimation, our model, ProximalFM, explicitly targets the Bayesian posterior distribution of the CATE. One unique aspect of this problem is that we need to provide Monte Carlo estimates of the oracle CATEs, leading to a novel variation of PFNs that accounts for the added stochastic error. ProximalFM is competitive with existing estimators across a diverse set of proximal regimes, with its clearest advantage under substantial latent confounding and informative proxies. It attains this performance without dataset-specific tuning, in a single amortized forward pass.