# Approximating Counterfactual Bounds while Fusing Observational, Biased and Randomised Data Sources

arXiv: https://arxiv.org/abs/2307.16577 · HTML: https://mlsanity.com/paper/2307.16577/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 23 references that are in the map.
- First posted on arXiv 31 Jul 2023.

## Builds on (most cited first, 9 of 9)

- [Probabilities of Causation: Bounds and Identification](https://mlsanity.com/paper/1301.3898/) (2013)
- [Counterfactual Probabilities: Computational Methods, Bounds and Applications](https://mlsanity.com/paper/1302.6784/) (2013)
- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [An Automated Approach to Causal Inference in Discrete Settings](https://mlsanity.com/paper/2109.13471/) (2021)
- [Personalized Decision Making -- A Conceptual Introduction](https://mlsanity.com/paper/2208.09558/) (2022)
- [Structural Causal Models Are (Solvable by) Credal Networks](https://mlsanity.com/paper/2008.00463/) (2020)
- [Causal Inference Using Tractable Circuits](https://mlsanity.com/paper/2202.02891/) (2022)
- [Causal Expectation-Maximisation](https://mlsanity.com/paper/2011.02912/) (2020)
- [Bounding Counterfactuals under Selection Bias](https://mlsanity.com/paper/2208.01417/) (2022)

## Built on by (most cited first, 9 of 9)

- [Zero-shot Causal Graph Extrapolation from Text via LLMs](https://mlsanity.com/paper/2312.14670/) (2023)
- [Integral Imprecise Probability Metrics](https://mlsanity.com/paper/2505.16156/) (2025)
- [Beyond Prediction-Set Size: Quantifying Conformal Uncertainty through Credal Geometry](https://mlsanity.com/paper/2602.01667/) (2026)
- [Identifying Causal Effects Under Functional Dependencies](https://mlsanity.com/paper/2403.04919/) (2024)
- [Robust Counterfactual Inference in Markov Decision Processes](https://mlsanity.com/paper/2502.13731/) (2025)
- [Bayesian Networks for Causal Analysis in Socioecological Systems](https://mlsanity.com/paper/2401.10101/) (2024)
- [When Do Credal Sets Stabilize? Fixed-Point Theorems for Credal Set Updates](https://mlsanity.com/paper/2510.04769/) (2025)
- [Partial Identification under Causal Orders by Linear Programming](https://mlsanity.com/paper/2608.24427/) (2026)
- [Causal Modelling of Support Interventions for Student Competency Assessment](https://mlsanity.com/paper/2608.24632/) (2026)

## Abstract (arXiv)

We address the problem of integrating data from multiple, possibly biased, observational and interventional studies, to eventually compute counterfactuals in structural causal models. We start from the case of a single observational dataset affected by a selection bias. We show that the likelihood of the available data has no local maxima. This enables us to use the causal expectation-maximisation scheme to approximate the bounds for partially identifiable counterfactual queries, which are the focus of this paper. We then show how the same approach can address the general case of multiple datasets, no matter whether interventional or observational, biased or unbiased, by remapping it into the former one via graphical transformations. Systematic numerical experiments and a case study on palliative care show the effectiveness of our approach, while hinting at the benefits of fusing heterogeneous data sources to get informative outcomes in case of partial identifiability.