# Learning Robust Statistics for Simulation-based Inference under Model Misspecification

arXiv: https://arxiv.org/abs/2305.15871 · HTML: https://mlsanity.com/paper/2305.15871/

## Key facts

- Cited by 48 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 87 references that are in the map.
- First posted on arXiv 25 May 2023.

## Builds on (most cited first, 30 of 53)

- [Deep Sets](https://mlsanity.com/paper/1703.06114/) (2017)
- [Masked Autoregressive Flow for Density Estimation](https://mlsanity.com/paper/1705.07057/) (2017)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Kernel Mean Embedding of Distributions: A Review and Beyond](https://mlsanity.com/paper/1605.09522/) (2016)
- [Sequential Neural Likelihood: Fast Likelihood-free Inference with Autoregressive Flows](https://mlsanity.com/paper/1805.07226/) (2018)
- [Automatic Posterior Transformation for Likelihood-Free Inference](https://mlsanity.com/paper/1905.07488/) (2019)
- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [BayesFlow: Learning complex stochastic models with invertible neural networks](https://mlsanity.com/paper/2003.06281/) (2020)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Neural Approximate Sufficient Statistics for Implicit Models](https://mlsanity.com/paper/2010.10079/) (2020)
- [Likelihood-free inference with emulator networks](https://mlsanity.com/paper/1805.09294/) (2018)
- [Generalized Variational Inference: Three arguments for deriving new Posteriors](https://mlsanity.com/paper/1904.02063/) (2019)
- [Sequential Neural Score Estimation: Likelihood-Free Inference with Conditional Score Based Diffusion Models](https://mlsanity.com/paper/2210.04872/) (2022)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks](https://mlsanity.com/paper/2112.08866/) (2021)
- [Robust Neural Posterior Estimation and Statistical Model Criticism](https://mlsanity.com/paper/2210.06564/) (2022)
- [Statistical Inference for Generative Models with Maximum Mean Discrepancy](https://mlsanity.com/paper/1906.05944/) (2019)
- [Investigating the Impact of Model Misspecification in Neural Simulation-based Inference](https://mlsanity.com/paper/2209.01845/) (2022)
- [Variational methods for simulation-based inference](https://mlsanity.com/paper/2203.04176/) (2022)
- [Robust Bayesian Inference for Simulator-based Models via the MMD Posterior Bootstrap](https://mlsanity.com/paper/2202.04744/) (2022)
- [Learning Summary Statistic for Approximate Bayesian Computation via Deep Neural Network](https://mlsanity.com/paper/1510.02175/) (2015)
- [Conditional Density Estimation with Neural Networks: Best Practices and Benchmarks](https://mlsanity.com/paper/1903.00954/) (2019)
- [GATSBI: Generative Adversarial Training for Simulation-Based Inference](https://mlsanity.com/paper/2203.06481/) (2022)
- [MMD-Bayes: Robust Bayesian Estimation via Maximum Mean Discrepancy](https://mlsanity.com/paper/1909.13339/) (2019)
- [A Likelihood-Free Inference Framework for Population Genetic Data using Exchangeable Neural Networks](https://mlsanity.com/paper/1802.06153/) (2018)
- [Sequential Neural Posterior and Likelihood Approximation](https://mlsanity.com/paper/2102.06522/) (2021)
- [Bayesian Synthetic Likelihood](https://mlsanity.com/paper/2305.05120/) (2023)
- [Towards Reliable Simulation-Based Inference with Balanced Neural Ratio Estimation](https://mlsanity.com/paper/2208.13624/) (2022)
- [Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence](https://mlsanity.com/paper/1912.05737/) (2019)

## Built on by (most cited first, 30 of 48)

- [Simulation-Based Inference: A Practical Guide](https://mlsanity.com/paper/2508.12939/) (2025)
- [Addressing Misspecification in Simulation-based Inference through Data-driven Calibration](https://mlsanity.com/paper/2405.08719/) (2024)
- [Amortized Probabilistic Conditioning for Optimization, Simulation and Inference](https://mlsanity.com/paper/2410.15320/) (2024)
- [Amortized Bayesian Experimental Design for Decision-Making](https://mlsanity.com/paper/2411.02064/) (2024)
- [Sensitivity-Aware Amortized Bayesian Inference](https://mlsanity.com/paper/2310.11122/) (2023)
- [Amortized Bayesian Multilevel Models](https://mlsanity.com/paper/2408.13230/) (2024)
- [Robust Amortized Bayesian Inference with Self-Consistency Losses on Unlabeled Data](https://mlsanity.com/paper/2501.13483/) (2025)
- [Solving Empirical Bayes via Transformers](https://mlsanity.com/paper/2502.09844/) (2025)
- [Does Unsupervised Domain Adaptation Improve the Robustness of Amortized Bayesian Inference? A Systematic Evaluation](https://mlsanity.com/paper/2502.04949/) (2025)
- [Multifidelity Simulation-based Inference for Computationally Expensive Simulators](https://mlsanity.com/paper/2502.08416/) (2025)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)
- [A Dictionary of Closed-Form Kernel Mean Embeddings](https://mlsanity.com/paper/2504.18830/) (2025)
- [BayesFlow 2: Multi-Backend Amortized Bayesian Inference in Python](https://mlsanity.com/paper/2602.07098/) (2026)
- [Detecting Model Misspecification in Amortized Bayesian Inference with Neural Networks: An Extended Investigation](https://mlsanity.com/paper/2406.03154/) (2024)
- [ALINE: Joint Amortization for Bayesian Inference and Active Data Acquisition](https://mlsanity.com/paper/2506.07259/) (2025)
- [Preconditioned Neural Posterior Estimation for Likelihood-free Inference](https://mlsanity.com/paper/2404.13557/) (2024)
- [A Comprehensive Guide to Simulation-based Inference in Computational Biology](https://mlsanity.com/paper/2409.19675/) (2024)
- [Cost-aware simulation-based inference](https://mlsanity.com/paper/2410.07930/) (2024)
- [Robust Simulation-Based Inference under Missing Data via Neural Processes](https://mlsanity.com/paper/2503.01287/) (2025)
- [Deep Calibration of Market Simulations using Neural Density Estimators and Embedding Networks](https://mlsanity.com/paper/2311.11913/) (2023)
- [Inductive Domain Transfer In Misspecified Simulation-Based Inference](https://mlsanity.com/paper/2508.15593/) (2025)
- [Fuse It or Lose It: Deep Fusion for Multimodal Simulation-Based Inference](https://mlsanity.com/paper/2311.10671/) (2023)
- [A Quadrature Approach for General-Purpose Batch Bayesian Optimization via Probabilistic Lifting](https://mlsanity.com/paper/2404.12219/) (2024)
- [Amortized Bayesian Workflow](https://mlsanity.com/paper/2409.04332/) (2024)
- [Posterior SBC: Simulation-Based Calibration Checking Conditional on Data](https://mlsanity.com/paper/2502.03279/) (2025)
- [PriorGuide: Test-Time Prior Adaptation for Simulation-Based Inference](https://mlsanity.com/paper/2510.13763/) (2025)
- [Minimum Distance Summaries for Robust Neural Posterior Estimation](https://mlsanity.com/paper/2602.09161/) (2026)
- [Cryo-em images are intrinsically low dimensional](https://mlsanity.com/paper/2504.11249/) (2025)
- [Representative, Informative, and De-Amplifying: Requirements for Robust Bayesian Active Learning under Model Misspecification](https://mlsanity.com/paper/2506.07805/) (2025)
- [Flow Matching Calibration for Simulation-Based Inference under Model Misspecification](https://mlsanity.com/paper/2509.23385/) (2025)

## Abstract (arXiv)

Simulation-based inference (SBI) methods such as approximate Bayesian computation (ABC), synthetic likelihood, and neural posterior estimation (NPE) rely on simulating statistics to infer parameters of intractable likelihood models. However, such methods are known to yield untrustworthy and misleading inference outcomes under model misspecification, thus hindering their widespread applicability. In this work, we propose the first general approach to handle model misspecification that works across different classes of SBI methods. Leveraging the fact that the choice of statistics determines the degree of misspecification in SBI, we introduce a regularized loss function that penalises those statistics that increase the mismatch between the data and the model. Taking NPE and ABC as use cases, we demonstrate the superior performance of our method on high-dimensional time-series models that are artificially misspecified. We also apply our method to real data from the field of radio propagation where the model is known to be misspecified. We show empirically that the method yields robust inference in misspecified scenarios, whilst still being accurate when the model is well-specified.