# An Automated Approach to Causal Inference in Discrete Settings

arXiv: https://arxiv.org/abs/2109.13471 · HTML: https://mlsanity.com/paper/2109.13471/

## Key facts

- Cited by 42 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 63 references that are in the map.
- First posted on arXiv 28 Sep 2021.

## Builds on (most cited first, 16 of 16)

- [Variational Inference: A Review for Statisticians](https://mlsanity.com/paper/1601.00670/) (2016)
- [Counterfactual Probabilities: Computational Methods, Bounds and Applications](https://mlsanity.com/paper/1302.6784/) (2013)
- [Nested Markov Properties for Acyclic Directed Mixed Graphs](https://mlsanity.com/paper/1701.06686/) (2017)
- [On the Testability of Causal Models with Latent and Instrumental Variables](https://mlsanity.com/paper/1302.4976/) (2013)
- [Margins of discrete Bayesian networks](https://mlsanity.com/paper/1501.02103/) (2015)
- [Bounds on Causal Effects and Application to High Dimensional Data](https://mlsanity.com/paper/2106.12121/) (2021)
- [A General Method for Deriving Tight Symbolic Bounds on Causal Effects](https://mlsanity.com/paper/2003.10702/) (2020)
- [The Inflation Technique for Causal Inference with Latent Variables](https://mlsanity.com/paper/1609.00672/) (2016)
- [Partial Identifiability in Discrete Data With Measurement Error](https://mlsanity.com/paper/2012.12449/) (2020)
- [Instrumentality Tests Revisited](https://mlsanity.com/paper/1301.2258/) (2013)
- [Quantifier Elimination for Statistical Problems](https://mlsanity.com/paper/1301.6698/) (2013)
- [Smooth, identifiable supermodels of discrete DAG models with latent variables](https://mlsanity.com/paper/1511.06813/) (2015)
- [The HulC: Confidence Regions from Convex Hulls](https://mlsanity.com/paper/2105.14577/) (2021)
- [Survivor-complier effects in the presence of selection on treatment, with application to a study of prompt ICU admission](https://mlsanity.com/paper/1704.05706/) (2017)
- [Optimal Confidence Regions for the Multinomial Parameter](https://mlsanity.com/paper/2002.01044/) (2020)
- [Causal bounds for outcome-dependent sampling in observational studies](https://mlsanity.com/paper/2002.10519/) (2020)

## Built on by (most cited first, 30 of 42)

- [Stochastic Causal Programming for Bounding Treatment Effects](https://mlsanity.com/paper/2202.10806/) (2022)
- [Sharp Bounds for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2305.16988/) (2023)
- [Data-Driven Causal Effect Estimation Based on Graphical Causal Modelling: A Survey](https://mlsanity.com/paper/2208.09590/) (2022)
- [Partial Identification of Treatment Effects with Implicit Generative Models](https://mlsanity.com/paper/2210.08139/) (2022)
- [A Neural Framework for Generalized Causal Sensitivity Analysis](https://mlsanity.com/paper/2311.16026/) (2023)
- [Partial Identification with Noisy Covariates: A Robust Optimization Approach](https://mlsanity.com/paper/2202.10665/) (2022)
- [Meta-Learners for Partially-Identified Treatment Effects Across Multiple Environments](https://mlsanity.com/paper/2406.02464/) (2024)
- [Falsification before Extrapolation in Causal Effect Estimation](https://mlsanity.com/paper/2209.13708/) (2022)
- [IGC-Net for conditional average potential outcome estimation over time](https://mlsanity.com/paper/2405.21012/) (2024)
- [Approximating Counterfactual Bounds while Fusing Observational, Biased and Randomised Data Sources](https://mlsanity.com/paper/2307.16577/) (2023)
- [Efficient Computation of Counterfactual Bounds](https://mlsanity.com/paper/2307.08304/) (2023)
- [The Fragility of Fairness: Causal Sensitivity Analysis for Fair Machine Learning](https://mlsanity.com/paper/2410.09600/) (2024)
- [Causal Discovery in Linear Latent Variable Models Subject to Measurement Error](https://mlsanity.com/paper/2211.03984/) (2022)
- [On the Complexity of Counterfactual Reasoning](https://mlsanity.com/paper/2211.13447/) (2022)
- [Constructing Confidence Intervals for Average Treatment Effects from Multiple Datasets](https://mlsanity.com/paper/2412.11511/) (2024)
- [Scalable Computation of Causal Bounds](https://mlsanity.com/paper/2308.02709/) (2023)
- [Consistency of Neural Causal Partial Identification](https://mlsanity.com/paper/2405.15673/) (2024)
- [Learning Representations of Instruments for Partial Identification of Treatment Effects](https://mlsanity.com/paper/2410.08976/) (2024)
- [The Causal Marginal Polytope for Bounding Treatment Effects](https://mlsanity.com/paper/2202.13851/) (2022)
- [Bounding probabilities of causation through the causal marginal problem](https://mlsanity.com/paper/2304.02023/) (2023)
- [Robust Counterfactual Inference in Markov Decision Processes](https://mlsanity.com/paper/2502.13731/) (2025)
- [IV-ICL: Bounding Causal Effects with Instrumental Variables via In-Context Learning](https://mlsanity.com/paper/2605.12924/) (2026)
- [Tractable Bounding of Counterfactual Queries by Knowledge Compilation](https://mlsanity.com/paper/2310.03352/) (2023)
- [Multilinear and Linear Programs for Partially Identifiable Queries in Quasi-Markovian Structural Causal Models](https://mlsanity.com/paper/2509.03548/) (2025)
- [Foundation Models for Partial Causal Identification](https://mlsanity.com/paper/2608.20841/) (2026)
- [Tightening Bounds on Probabilities of Causation By Merging Datasets](https://mlsanity.com/paper/2310.08406/) (2023)
- [Your Assumed DAG is Wrong and Here's How To Deal With It](https://mlsanity.com/paper/2502.17030/) (2025)
- [Assessing the robustness of heterogeneous treatment effects in survival analysis under informative censoring](https://mlsanity.com/paper/2510.13397/) (2025)
- [Bounds and Identification of Joint Probabilities of Potential Outcomes and Observed Variables under Monotonicity Assumptions](https://mlsanity.com/paper/2602.18762/) (2026)
- [Learning to Bound Counterfactual Inference from Observational, Biased and Randomised Data](https://mlsanity.com/paper/2212.02932/) (2022)

## Abstract (arXiv)

When causal quantities cannot be point identified, researchers often pursue partial identification to quantify the range of possible values. However, the peculiarities of applied research conditions can make this analytically intractable. We present a general and automated approach to causal inference in discrete settings. We show causal questions with discrete data reduce to polynomial programming problems, and we present an algorithm to automatically bound causal effects using efficient dual relaxation and spatial branch-and-bound techniques. The user declares an estimand, states assumptions, and provides data (however incomplete or mismeasured). The algorithm then searches over admissible data-generating processes and outputs the most precise possible range consistent with available information -- i.e., sharp bounds -- including a point-identified solution if one exists. Because this search can be computationally intensive, our procedure reports and continually refines non-sharp ranges that are guaranteed to contain the truth at all times, even when the algorithm is not run to completion. Moreover, it offers an additional guarantee we refer to as ε-sharpness, characterizing the worst-case looseness of the incomplete bounds. Analytically validated simulations show the algorithm accommodates classic obstacles, including confounding, selection, measurement error, noncompliance, and nonresponse.