# CAFE: Counterfactual Prediction via Fast Posterior Estimation

arXiv: https://arxiv.org/abs/2609.32167 · HTML: https://mlsanity.com/paper/2609.32167/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 32 of its 46 references that are in the map.
- First posted on arXiv 26 Sep 2026.

## Builds on (most cited first, 30 of 32)

- [Causal Discovery with Continuous Additive Noise Models](https://mlsanity.com/paper/1309.6779/) (2013)
- [Deep Structural Causal Models for Tractable Counterfactual Inference](https://mlsanity.com/paper/2006.06485/) (2020)
- [Algorithmic recourse under imperfect causal knowledge: a probabilistic approach](https://mlsanity.com/paper/2006.06831/) (2020)
- [TabPFN-2.5: Advancing the State of the Art in Tabular Foundation Models](https://mlsanity.com/paper/2511.08667/) (2025)
- [Counterfactual Off-Policy Evaluation with Gumbel-Max Structural Causal Models](https://mlsanity.com/paper/1905.05824/) (2019)
- [DiBS: Differentiable Bayesian Structure Learning](https://mlsanity.com/paper/2105.11839/) (2021)
- [Causal Autoregressive Flows](https://mlsanity.com/paper/2011.02268/) (2020)
- [Deep End-to-end Causal Inference](https://mlsanity.com/paper/2202.02195/) (2022)
- [Learning Counterfactual Representations for Estimating Individual Dose-Response Curves](https://mlsanity.com/paper/1902.00981/) (2019)
- [LimiX: Unleashing Structured-Data Modeling Capability for Generalist Intelligence](https://mlsanity.com/paper/2509.03505/) (2025)
- [Estimating the Effects of Continuous-valued Interventions using Generative Adversarial Networks](https://mlsanity.com/paper/2002.12326/) (2020)
- [Causal normalizing flows: from theory to practice](https://mlsanity.com/paper/2306.05415/) (2023)
- [Do-PFN: In-Context Learning for Causal Effect Estimation](https://mlsanity.com/paper/2506.06039/) (2025)
- [CausalPFN: Amortized Causal Effect Estimation via In-Context Learning](https://mlsanity.com/paper/2506.07918/) (2025)
- [Foundation Models for Causal Inference via Prior-Data Fitted Networks](https://mlsanity.com/paper/2506.10914/) (2025)
- [Modeling Causal Mechanisms with Diffusion Models for Interventional and Counterfactual Queries](https://mlsanity.com/paper/2302.00860/) (2023)
- [Self-Compatibility: Evaluating Causal Discovery without Ground Truth](https://mlsanity.com/paper/2307.09552/) (2023)
- [Counterfactual Explanations as Interventions in Latent Space](https://mlsanity.com/paper/2106.07754/) (2021)
- [Counterfactual (Non-)identifiability of Learned Structural Causal Models](https://mlsanity.com/paper/2301.09031/) (2023)
- [Estimating Interventional Distributions with Uncertain Causal Graphs through Meta-Learning](https://mlsanity.com/paper/2507.05526/) (2025)
- [Counterfactual Generative Modeling with Variational Causal Inference](https://mlsanity.com/paper/2410.12730/) (2024)
- [A Fixed-Point Approach for Causal Generative Modeling](https://mlsanity.com/paper/2404.06969/) (2024)
- [Use What You Know: Causal Foundation Models with Partial Graphs](https://mlsanity.com/paper/2602.14972/) (2026)
- [Using proper divergence functions to evaluate climate models](https://mlsanity.com/paper/1301.5927/) (2013)
- [Hybrid Top-Down Global Causal Discovery with Local Search for Linear and Nonlinear Additive Noise Models](https://mlsanity.com/paper/2405.14496/) (2024)
- [Amortized Inference of Causal Models via Conditional Fixed-Point Iterations](https://mlsanity.com/paper/2410.06128/) (2024)
- [CausalMan: A physics-based simulator for large-scale causality](https://mlsanity.com/paper/2502.12707/) (2025)
- [Causal Foundation Models with Continuous Treatments](https://mlsanity.com/paper/2605.15133/) (2026)
- [Causality on Cross-Sectional Data: Stable Specification Search in Constrained Structural Equation Modeling](https://mlsanity.com/paper/1506.05600/) (2015)
- [A Causal Foundation Model for Structure and Outcome Prediction](https://mlsanity.com/paper/2606.26467/) (2026)

## Abstract (arXiv)

Counterfactual prediction estimates an individual's outcome under an alternative intervention given their factual observations. Such outcomes are generally not identifiable from observational data without additional assumptions. Even within the class of fully observed additive noise models (ANMs), different causal graphs can generate the same observational distribution yet imply different individual counterfactual outcomes. Predictions based on a single estimated graph ignore this structural uncertainty. We therefore target a Bayesian counterfactual posterior predictive distribution that combines predictions from plausible SCMs. We introduce CAFE (CounterfActual Prediction via Fast Posterior Estimation), an amortized inference framework that directly approximates the Bayesian counterfactual posterior predictive distribution. We pretrain a transformer-based model on synthetic counterfactual tasks generated from a diverse prior over ANMs. Given an observational dataset, an individual's factual observations, and an intervention, CAFE approximates the corresponding posterior predictive distribution in a single forward pass. Experiments show that CAFE accurately predicts individual counterfactual outcomes in identifiable settings and approximates the posterior predictive distribution when structural uncertainty induced by observationally indistinguishable causal graphs exists. Strong performance in realistic manufacturing and viticulture settings further demonstrates its empirical robustness beyond the assumptions of the training prior.