# Validated Variational Inference via Practical Posterior Error Bounds

arXiv: https://arxiv.org/abs/1910.04102 · HTML: https://mlsanity.com/paper/1910.04102/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Oct 2019.

## Built on by (most cited first, 26 of 26)

- [Challenges and Opportunities in High-dimensional Variational Inference](https://mlsanity.com/paper/2103.01085/) (2021)
- [Variational Refinement for Importance Sampling Using the Forward Kullback-Leibler Divergence](https://mlsanity.com/paper/2106.15980/) (2021)
- [Black Box Variational Inference with a Deterministic Objective: Faster, More Accurate, and Even More Black Box](https://mlsanity.com/paper/2304.05527/) (2023)
- [Variational Inference for Uncertainty Quantification: an Analysis of Trade-offs](https://mlsanity.com/paper/2403.13748/) (2024)
- [Variational Inference in Location-Scale Families: Exact Recovery of the Mean and Correlation Matrix](https://mlsanity.com/paper/2410.11067/) (2024)
- [The computational asymptotics of Gaussian variational inference and the Laplace approximation](https://mlsanity.com/paper/2104.05886/) (2021)
- [Toward a Taxonomy of Trust for Probabilistic Machine Learning](https://mlsanity.com/paper/2112.03270/) (2021)
- [Are you using test log-likelihood correctly?](https://mlsanity.com/paper/2212.00219/) (2022)
- [Statistical and Computational Trade-offs in Variational Inference: A Case Study in Inferential Model Selection](https://mlsanity.com/paper/2207.11208/) (2022)
- [Measuring the robustness of Gaussian processes to kernel choice](https://mlsanity.com/paper/2106.06510/) (2021)
- [Many processors, little time: MCMC for partitions via optimal transport couplings](https://mlsanity.com/paper/2202.11258/) (2022)
- [Bounding Wasserstein distance with couplings](https://mlsanity.com/paper/2112.03152/) (2021)
- [Bernstein Flows for Flexible Posteriors in Variational Bayes](https://mlsanity.com/paper/2202.05650/) (2022)
- [Generalized Guarantees for Variational Inference in the Presence of Even and Elliptical Symmetry](https://mlsanity.com/paper/2511.01064/) (2025)
- [SOAP-Bubbles: Structured Weight Uncertainty for Neural Networks](https://mlsanity.com/paper/2606.23357/) (2026)
- [Discovering Inductive Bias with Gibbs Priors: A Diagnostic Tool for Approximate Bayesian Inference](https://mlsanity.com/paper/2203.03353/) (2022)
- [Centered plug-in estimation of Wasserstein distances](https://mlsanity.com/paper/2203.11627/) (2022)
- [Variational excess risk bound for general state space models](https://mlsanity.com/paper/2312.09607/) (2023)
- [Recommendations for Baselines and Benchmarking Approximate Gaussian Processes](https://mlsanity.com/paper/2402.09849/) (2024)
- [Budget-Constrained Bounds for Mini-Batch Estimation of Optimal Transport](https://mlsanity.com/paper/2210.13630/) (2022)
- [Variational Bayesian Neural Networks via Resolution of Singularities](https://mlsanity.com/paper/2302.06035/) (2023)
- [A Targeted Accuracy Diagnostic for Variational Approximations](https://mlsanity.com/paper/2302.12419/) (2023)
- [Maxitive Donsker-Varadhan Formulation for Possibilistic Variational Inference](https://mlsanity.com/paper/2511.21223/) (2025)
- [Even More Guarantees for Variational Inference in the Presence of Symmetries](https://mlsanity.com/paper/2604.21407/) (2026)
- [Accurate Large-sample Uncertainty Quantification using Stochastic Gradient Markov Chain Monte Carlo](https://mlsanity.com/paper/2606.00293/) (2026)
- [Large-scale empirical tuning and comparison of default optimizers for variational inference](https://mlsanity.com/paper/2606.07841/) (2026)

## Abstract (arXiv)

Variational inference has become an increasingly attractive fast alternative to Markov chain Monte Carlo methods for approximate Bayesian inference. However, a major obstacle to the widespread use of variational methods is the lack of post-hoc accuracy measures that are both theoretically justified and computationally efficient. In this paper, we provide rigorous bounds on the error of posterior mean and uncertainty estimates that arise from full-distribution approximations, as in variational inference. Our bounds are widely applicable, as they require only that the approximating and exact posteriors have polynomial moments. Our bounds are also computationally efficient for variational inference because they require only standard values from variational objectives, straightforward analytic calculations, and simple Monte Carlo estimates. We show that our analysis naturally leads to a new and improved workflow for validated variational inference. Finally, we demonstrate the utility of our proposed workflow and error bounds on a robust regression problem and on a real-data example with a widely used multilevel hierarchical model.