# Bridging Bayesian and Minimax Mean Square Error Estimation via Wasserstein Distributionally Robust Optimization

arXiv: https://arxiv.org/abs/1911.03539 · HTML: https://mlsanity.com/paper/1911.03539/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Nov 2019.

## Built on by (most cited first, 11 of 11)

- [Statistical Analysis of Wasserstein Distributionally Robust Estimators](https://mlsanity.com/paper/2108.02120/) (2021)
- [Counterfactual Plans under Distributional Ambiguity](https://mlsanity.com/paper/2201.12487/) (2022)
- [Outlier-Robust Wasserstein DRO](https://mlsanity.com/paper/2311.05573/) (2023)
- [Distributionally Robust Fair Principal Components via Geodesic Descents](https://mlsanity.com/paper/2202.03071/) (2022)
- [Nash Equilibria, Regularization and Computation in Optimal Transport-Based Distributionally Robust Optimization](https://mlsanity.com/paper/2303.03900/) (2023)
- [Adversarial Regression with Doubly Non-negative Weighting Matrices](https://mlsanity.com/paper/2109.14875/) (2021)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [A Geometric Unification of Distributionally Robust Covariance Estimators: Shrinking the Spectrum by Inflating the Ambiguity Set](https://mlsanity.com/paper/2405.20124/) (2024)
- [Nonlinear Distributionally Robust Optimization](https://mlsanity.com/paper/2306.03202/) (2023)
- [Wasserstein Distributionally Robust Risk-Sensitive Estimation via Conditional Value-at-Risk](https://mlsanity.com/paper/2604.18546/) (2026)
- [A Distributionally Robust Framework for Learned Reconstructions in Inverse Problems](https://mlsanity.com/paper/2606.30230/) (2026)

## Abstract (arXiv)

We introduce a distributionally robust minimium mean square error estimation model with a Wasserstein ambiguity set to recover an unknown signal from a noisy observation. The proposed model can be viewed as a zero-sum game between a statistician choosing an estimator -- that is, a measurable function of the observation -- and a fictitious adversary choosing a prior -- that is, a pair of signal and noise distributions ranging over independent Wasserstein balls -- with the goal to minimize and maximize the expected squared estimation error, respectively. We show that if the Wasserstein balls are centered at normal distributions, then the zero-sum game admits a Nash equilibrium, where the players' optimal strategies are given by an em affine estimator and a em normal prior, respectively. We further prove that this Nash equilibrium can be computed by solving a tractable convex program. Finally, we develop a Frank-Wolfe algorithm that can solve this convex program orders of magnitude faster than state-of-the-art general purpose solvers. We show that this algorithm enjoys a linear convergence rate and that its direction-finding subproblems can be solved in quasi-closed form.