# Shrinkage priors for Bayesian Substitute Confounders

arXiv: https://arxiv.org/abs/2606.18535 · HTML: https://mlsanity.com/paper/2606.18535/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 41 references that are in the map.
- First posted on arXiv 16 Jun 2026.

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

- [Estimation and Inference of Heterogeneous Treatment Effects using Random Forests](https://mlsanity.com/paper/1510.04342/) (2015)
- [Estimating Counterfactual Treatment Outcomes over Time Through Adversarially Balanced Representations](https://mlsanity.com/paper/2002.04083/) (2020)
- [Identifying Causal Effects With Proxy Variables of an Unmeasured Confounder](https://mlsanity.com/paper/1609.08816/) (2016)
- [The Blessings of Multiple Causes](https://mlsanity.com/paper/1805.06826/) (2018)
- [An Introduction to Proximal Causal Learning](https://mlsanity.com/paper/2009.10982/) (2020)
- [Robust Inference on Average Treatment Effects with Possibly More Covariates than Observations](https://mlsanity.com/paper/1309.4686/) (2013)
- [Using Embeddings to Correct for Unobserved Confounding in Networks](https://mlsanity.com/paper/1902.04114/) (2019)
- [Estimation of causal effects with multiple treatments: a review and new ideas](https://mlsanity.com/paper/1701.05132/) (2017)
- [Augmented balancing weights as linear regression](https://mlsanity.com/paper/2304.14545/) (2023)
- [On Multi-Cause Causal Inference with Unobserved Confounding: Counterexamples, Impossibility, and Alternatives](https://mlsanity.com/paper/1902.10286/) (2019)
- [Comment on "Blessings of Multiple Causes"](https://mlsanity.com/paper/1910.05438/) (2019)
- [Multiple Causal Inference with Latent Confounding](https://mlsanity.com/paper/1805.08273/) (2018)
- [Bayesian cumulative shrinkage for infinite factorizations](https://mlsanity.com/paper/1902.04349/) (2019)
- [Implicit Causal Models for Genome-wide Association Studies](https://mlsanity.com/paper/1710.10742/) (2017)
- [Spatial Deconfounder: Interference-Aware Deconfounding for Spatial Causal Inference](https://mlsanity.com/paper/2510.08762/) (2025)
- [Wedge Sampling: Efficient Tensor Completion with Nearly-Linear Sample Complexity](https://mlsanity.com/paper/2602.05869/) (2026)

## Abstract (arXiv)

Multi-cause observational studies contain information about unmeasured confounding through the dependence structure among causes. However, literal imputation of the unobserved confounder is often more complex than learning a lower-dimensional substitute score that preserves the shared assignment variation needed for stable causal adjustment. The deconfounder (Wang and Blei, 2019) and related substitute confounder methods exploit this idea, but flexible assignment models can fit the joint distribution of the causes while producing scores that over-encode the treatment vector, collapse overlap, or capture single-cause variation. We develop a Bayesian factor assignment framework for learning sparse substitute confounders that retain coarse multi-cause dependence with shrinkage priors. The theory is stated at the level of posterior concentration, factor score contraction, and overlap-preserving assignment geometry and therefore does not rely on a particular shrinkage prior. Under these conditions, the proposed regression-adjusted estimators are consistent for mean potential outcomes when the corresponding latent variable identification assumptions hold. Shrinkage priors provide a natural tool for latent structural learning: they favour low-dimensional factors supported by multiple causes, discourage effectively single-cause factors, and induce an ordering of the latent factors through progressive shrinkage. Synthetic experiments illustrate the roles of signal strength, outcome validity, and geometry-aware regularization. In an Alzheimer's Disease Neuroimaging Initiative (ADNI) baseline analysis, sparse substitute scores recover much of the adjustment obtained by directly conditioning on invasive cerebrospinal-fluid biomarkers, while collapse diagnostics identify when fitted factors reduce to individual observed measurements.