# Robust Bayesian Inference for Measurement Error Misspecification: The Berkson and Classical Cases

arXiv: https://arxiv.org/abs/2306.01468 · HTML: https://mlsanity.com/paper/2306.01468/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 65 references that are in the map.
- First posted on arXiv 2 Jun 2023.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Characteristic and Universal Tensor Product Kernels](https://mlsanity.com/paper/1708.08157/) (2017)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [A Rigorous Theory of Conditional Mean Embeddings](https://mlsanity.com/paper/1912.00671/) (2019)
- [Nonparametric learning from Bayesian models with randomized objective functions](https://mlsanity.com/paper/1806.11544/) (2018)
- [Scalable Nonparametric Sampling from Multimodal Posteriors with the Posterior Bootstrap](https://mlsanity.com/paper/1902.03175/) (2019)
- [Closed-form Expressions for Maximum Mean Discrepancy with Applications to Wasserstein Auto-Encoders](https://mlsanity.com/paper/1901.03227/) (2019)
- [Partial Identifiability in Discrete Data With Measurement Error](https://mlsanity.com/paper/2012.12449/) (2020)
- [Causal Inference with Treatment Measurement Error: A Nonparametric Instrumental Variable Approach](https://mlsanity.com/paper/2206.09186/) (2022)
- [Measurement error models: from nonparametric methods to deep neural networks](https://mlsanity.com/paper/2007.07498/) (2020)
- [Learning Models from Data with Measurement Error: Tackling Underreporting](https://mlsanity.com/paper/1901.09060/) (2019)
- [Convex Total Least Squares](https://mlsanity.com/paper/1406.0189/) (2014)
- [Total Least Squares Regression in Input Sparsity Time](https://mlsanity.com/paper/1909.12441/) (2019)

## Built on by (most cited first, 2 of 2)

- [On the Robustness of Kernel Goodness-of-Fit Tests](https://mlsanity.com/paper/2408.05854/) (2024)
- [A Gradient Flow Perspective on Minimum MMD Estimation](https://mlsanity.com/paper/2607.03871/) (2026)

## Abstract (arXiv)

Measurement error occurs when a covariate influencing a response variable is corrupted by noise. This can lead to misleading inference outcomes, particularly in problems where accurately estimating the relationship between covariates and response variables is crucial, such as causal effect estimation. Existing methods for dealing with measurement error often rely on strong assumptions such as knowledge of the error distribution or its variance and availability of replicated measurements of the covariates. We propose a Bayesian Nonparametric Learning framework that is robust to misspecification of these assumptions and does not require replicate measurements. This approach gives rise to a general framework that is suitable for both Classical and Berkson error models via the appropriate specification of the prior centering measure of a Dirichlet Process (DP). Moreover, it offers flexibility in the choice of loss function depending on the type of regression model. We provide bounds on the generalisation error based on the Maximum Mean Discrepancy (MMD) loss which allows for generalisation to non-Gaussian distributed errors and nonlinear covariate-response relationships. We showcase the effectiveness of the proposed framework versus prior art in real-world problems containing either Berkson or Classical measurement errors.