# BayesIMP: Uncertainty Quantification for Causal Data Fusion

arXiv: https://arxiv.org/abs/2106.03477 · HTML: https://mlsanity.com/paper/2106.03477/

## Key facts

- Cited by 21 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 32 references that are in the map.
- First posted on arXiv 7 Jun 2021.

## Builds on (most cited first, 13 of 13)

- [Aleatoric and Epistemic Uncertainty in Machine Learning: An Introduction to Concepts and Methods](https://mlsanity.com/paper/1910.09457/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Kernel Instrumental Variable Regression](https://mlsanity.com/paper/1906.00232/) (2019)
- [Causal Bayesian Optimization](https://mlsanity.com/paper/2005.11741/) (2020)
- [Large-Scale Kernel Methods for Independence Testing](https://mlsanity.com/paper/1606.07892/) (2016)
- [Counterfactual Mean Embeddings](https://mlsanity.com/paper/1805.08845/) (2018)
- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [Causal Inference via Kernel Deviance Measures](https://mlsanity.com/paper/1804.04622/) (2018)
- [Bayesian Learning of Kernel Embeddings](https://mlsanity.com/paper/1603.02160/) (2016)
- [Multi-task Causal Learning with Gaussian Processes](https://mlsanity.com/paper/2009.12821/) (2020)
- [Noise Contrastive Meta-Learning for Conditional Density Estimation using Kernel Mean Embeddings](https://mlsanity.com/paper/1906.02236/) (2019)
- [Bayesian Approaches to Distribution Regression](https://mlsanity.com/paper/1705.04293/) (2017)
- [Hyperparameter Learning for Conditional Kernel Mean Embeddings with Rademacher Complexity Bounds](https://mlsanity.com/paper/1809.00175/) (2018)

## Built on by (most cited first, 21 of 21)

- [Looping in the Human Collaborative and Explainable Bayesian Optimization](https://mlsanity.com/paper/2310.17273/) (2023)
- [Causal Inference Through the Structural Causal Marginal Problem](https://mlsanity.com/paper/2202.01300/) (2022)
- [Explaining the Uncertain: Stochastic Shapley Values for Gaussian Process Models](https://mlsanity.com/paper/2305.15167/) (2023)
- [GeneDisco: A Benchmark for Experimental Design in Drug Discovery](https://mlsanity.com/paper/2110.11875/) (2021)
- [Calibrating Multimodal Learning](https://mlsanity.com/paper/2306.01265/) (2023)
- [Domain Generalisation via Imprecise Learning](https://mlsanity.com/paper/2404.04669/) (2024)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Causal Strategic Learning with Competitive Selection](https://mlsanity.com/paper/2308.16262/) (2023)
- [Neural-Kernel Conditional Mean Embeddings](https://mlsanity.com/paper/2403.10859/) (2024)
- [An Overview of Causal Inference using Kernel Embeddings](https://mlsanity.com/paper/2410.22754/) (2024)
- [Exact Shapley Attributions in Quadratic-time for FANOVA Gaussian Processes](https://mlsanity.com/paper/2508.14499/) (2025)
- [Gated Domain Units for Multi-source Domain Generalization](https://mlsanity.com/paper/2206.12444/) (2022)
- [Causal Inference from Small High-dimensional Datasets](https://mlsanity.com/paper/2205.09281/) (2022)
- [Kernel-based estimators for functional causal effects](https://mlsanity.com/paper/2503.05024/) (2025)
- [Validation of Composite Systems by Discrepancy Propagation](https://mlsanity.com/paper/2210.12061/) (2022)
- [Interventional Processes for Causal Uncertainty Quantification](https://mlsanity.com/paper/2410.14483/) (2024)
- [Sequential Decision Making on Unmatched Data using Bayesian Kernel Embeddings](https://mlsanity.com/paper/2210.13692/) (2022)
- [ActiveCQ: Active Estimation of Causal Quantities](https://mlsanity.com/paper/2509.24293/) (2025)
- [Bayesian Counterfactual Mean Embeddings and Off-Policy Evaluation](https://mlsanity.com/paper/2211.01518/) (2022)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)
- [ShaplEIG: Bayesian Experimental Design for Shapley Value Estimation](https://mlsanity.com/paper/2606.02247/) (2026)

## Abstract (arXiv)

While causal models are becoming one of the mainstays of machine learning, the problem of uncertainty quantification in causal inference remains challenging. In this paper, we study the causal data fusion problem, where datasets pertaining to multiple causal graphs are combined to estimate the average treatment effect of a target variable. As data arises from multiple sources and can vary in quality and quantity, principled uncertainty quantification becomes essential. To that end, we introduce Bayesian Interventional Mean Processes, a framework which combines ideas from probabilistic integration and kernel mean embeddings to represent interventional distributions in the reproducing kernel Hilbert space, while taking into account the uncertainty within each causal graph. To demonstrate the utility of our uncertainty estimation, we apply our method to the Causal Bayesian Optimisation task and show improvements over state-of-the-art methods.