Causal Inference with Unobserved Confounding: A Mixture Learning Perspective

Mansi Sood; Devavrat Shah

cs.LGSep 19, 2026

Causal Inference with Unobserved Confounding: A Mixture Learning Perspective

Authors: Mansi Sood, Devavrat Shah

Organizations: Laboratory for Information and Decision Systems, Massachusetts Institute of Technology, Cambridge, MA 02139 USA

Abstract

Unobserved confounding is a fundamental challenge in causal inference from observational data. This article develops a mixture-learning perspective, viewing latent confounders as sources of heterogeneity that induce mixture structure in observed data. Under suitable structural and identifiability assumptions, recovering the mixing distribution and component mechanisms enables estimation of interventional distributions and causal estimands. Using variants of Bernoulli mixtures as a running example, we contextualize mixture-learning techniques and their structural assumptions, and connect them to causal inference in panel-data settings, including latent factor models and synthetic interventions.We then consider high-dimensional exponential-family mixtures with dependent outcome trajectories, moving beyond counterfactual means to model counterfactual distributions. We situate this perspective relative to complementary approaches for unobserved confounding. Together, these ideas provide a bridge between mixture learning and causal inference, connecting recent advances in high-dimensional mixture learning to scalable identification and estimation of causal effects while raising new challenges for mixture learning.

Figures & tables

Fig. 1: An illustration of a mixture-learning-based resolution to Simpson’s paradox. (a) The aggregate observational data (solid line) exhibits a spurious positive correlation between insulin dosage (

A

) and blood-sugar levels (

Y

). (b) When conditioned on this latent confounder (shown as orange and blue clusters), the true negative causal effect is revealed (dashed lines). (c) A mixture-learning perspective models unobserved confounding through a latent variable

U

that induces a mixture distribution. By learning the confounding density

p(u)

viewed as mixing density together with the component distributions

p(y\mid a,u)

, we can recover causal quantities from observational data under suitable identifiability conditions.

Fig. 2: Observational versus interventional worlds. While observational samples $(S^{(k)},D^{(k)})\sim p_{S,D}$ capture statistical associations between sickness ( $S$ ) and doctor visits ( $D$ ), they cannot determine interventional quantities, such as sickness outcomes under the intervention $\mathrm{do}(D=1)$ . This gap motivates the causal identifiability question: determining the structural assumptions required to learn interventional distributions from purely observational data.

Fig. 3: Evaluating candidate causal DAGs via intervention axioms. (Top) Observational data alone cannot distinguish between

G_{1}

(

S\to D

) and

G_{2}

(

S\leftarrow D

). (Middle) Under the intervention

\mathrm{do}(S=1)

, removing incoming edges to

S

correctly preserves

D\sim\mathrm{Ber}(0.8)

G_{1}

, but incorrectly isolates

D

G_{2}

. (Bottom) Under

\mathrm{do}(D=1)

, removing incoming edges to

D

correctly isolates the marginal

S\sim\mathrm{Ber}(0.5)

G_{1}

, while

G_{2}

incorrectly predicts

S\sim\mathrm{Ber}(0.8)

Fig. 4: From mixture learning to synthetic interventions: (a) A Bernoulli mixture ( 27 ) induces a low-rank matrix structure because each unit has a latent row-level parameter. (b) Synthetic interventions extend this idea to a low-rank unit, time, treatment tensor.

Fig. 5: Synthetic intervention view of unobserved confounding. Potential outcomes are modeled through unit, time, and treatment factors, inducing a low-rank tensor structure. Tensor completion uses the observed entries to estimate missing post-intervention counterfactual outcomes.

Fig. 6: Exponential-family model for treatment-outcome panels. Latent confounders induce dependencies among actions and outcomes, and the learned conditional distribution $f_{Y\mid U,A}$ supports counterfactual evaluation under alternative actions $\widetilde{a}$ .

Fig. 7: The canonical causal DAG (Fig. 1 ) contains an unobserved confounder $U$ affecting both treatment $A$ and outcome $Y$ . Randomized control trials externally assign $A$ to break its dependence on $U$ ; instrumental variables use an external variation $Z$ affecting $Y$ only through $A$ ; proximal learning uses observed proxies $Z$ and $W$ carrying information about $U$ ; and deconfounders leverage the multiplicity of causes to infer a latent substitute confounder by modeling the joint treatment distribution.

Mansi Sood ([email protected]) received the Ph.D. degree in Electrical and Computer Engineering from Carnegie Mellon University in 2024. She is a Schmidt Science Fellow at the MIT Laboratory for Information and Decision Systems. Her research focuses on network science, stochastic modeling, and inference. She received the A.G. Jordan Award for an outstanding Ph.D. thesis at CMU 2025, Information Theory and Applications Graduation Day Award 2024, Best Paper Award at the IEEE International Conference on Communications 2021, and EECS Rising Star awards 2021, 2023. She received the joint B.Tech. and M.Tech. degrees from IIT Bombay, where she was recognized for Excellence in Research and Mentorship.

Table 8

Devavrat Shah ([email protected]) (SM’16, Fellow’22) received the B.Tech. degree from IIT Bombay and the Ph.D. degree from Stanford University, both in computer science. He is currently the Andrew (1956) and Erna Viterbi Professor of Electrical Engineering and Computer Science at MIT. His research focuses on statistical inference, stochastic networks, and causal inference. Prof. Shah is the recipient of the 2026 ACM SIGMETRICS Achievement Award, 2024 INFORMS APS Markov Lecturer, 2010 INFORMS APS Erlang Prize, 2008 ACM SIGMETRICS Rising Star Award in addition to multiple paper awards and multiple Test of Time awards. He is a Kavli Fellow of the National Academy of Sciences, distinguished alumni of IIT Bombay and served as Editor-in-Chief of Stochastic Systems. He co-founded Celect (now Nike) and Ikigai Labs (now Celonis).

Table 9

Causal Inference with Unobserved Confounding

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We study causal inference under outcome interference for sequential, observational settings. Specifically, we consider settings where the binary outcomes over N units are Markovian across T time steps. At each time step, the outcomes of N units have dependencies captured through an Ising model; each outcome is also impacted through an external field capturing the effects of its treatment as well as latent confounders. Similar to panel data literature, these latent confounders are modeled to have a low-rank factor structure. Our data is a single sample from this high-dimensional distribution. To estimate causal quantities of interest, we provide a computationally efficient method based on Maximum Pseudo-Likelihood Estimation (MPLE) for learning the model parameters. Under mild assumptions, we establish non-asymptotic consistency for parameter estimation and show this translates to faithful estimation of causal quantities of interest after sampling from the learned model. We demonstrate the efficacy of the method through synthetic experiments as well as a real-world case-study investigating causal effects of vaccine rates on COVID-19 death rates within US counties nationwide.

Abstract

Causal Inference with Unobserved Confounding

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