# Causal Expectation-Maximisation

arXiv: https://arxiv.org/abs/2011.02912 · HTML: https://mlsanity.com/paper/2011.02912/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Nov 2020.

## Built on by (most cited first, 10 of 10)

- [Partial Counterfactual Identification from Observational and Experimental Data](https://mlsanity.com/paper/2110.05690/) (2021)
- [Approximating Counterfactual Bounds while Fusing Observational, Biased and Randomised Data Sources](https://mlsanity.com/paper/2307.16577/) (2023)
- [On the Complexity of Counterfactual Reasoning](https://mlsanity.com/paper/2211.13447/) (2022)
- [Bounding Counterfactuals under Selection Bias](https://mlsanity.com/paper/2208.01417/) (2022)
- [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)
- [Tractable Bounding of Counterfactual Queries by Knowledge Compilation](https://mlsanity.com/paper/2310.03352/) (2023)
- [Learning to Bound Counterfactual Inference from Observational, Biased and Randomised Data](https://mlsanity.com/paper/2212.02932/) (2022)
- [A Note on Bayesian Networks with Latent Root Variables](https://mlsanity.com/paper/2402.17087/) (2024)
- [Differentiable Expectation-Maximisation and Applications to Gaussian Mixture Model Optimal Transport](https://mlsanity.com/paper/2509.02109/) (2025)

## Abstract (arXiv)

Structural causal models are the basic modelling unit in Pearl's causal theory; in principle they allow us to solve counterfactuals, which are at the top rung of the ladder of causation. But they often contain latent variables that limit their application to special settings. This appears to be a consequence of the fact, proven in this paper, that causal inference is NP-hard even in models characterised by polytree-shaped graphs. To deal with such a hardness, we introduce the causal EM algorithm. Its primary aim is to reconstruct the uncertainty about the latent variables from data about categorical manifest variables. Counterfactual inference is then addressed via standard algorithms for Bayesian networks. The result is a general method to approximately compute counterfactuals, be they identifiable or not (in which case we deliver bounds). We show empirically, as well as by deriving credible intervals, that the approximation we provide becomes accurate in a fair number of EM runs. These results lead us finally to argue that there appears to be an unnoticed limitation to the trending idea that counterfactual bounds can often be computed without knowledge of the structural equations.