# Likelihood-Free Frequentist Inference: Bridging Classical Statistics and Machine Learning for Reliable Simulator-Based Inference

arXiv: https://arxiv.org/abs/2107.03920 · HTML: https://mlsanity.com/paper/2107.03920/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 94 references that are in the map.
- First posted on arXiv 8 Jul 2021.

## Builds on (most cited first, 27 of 27)

- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [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)
- [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)
- [Validating Bayesian Inference Algorithms with Simulation-Based Calibration](https://mlsanity.com/paper/1804.06788/) (2018)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [Approximating Likelihood Ratios with Calibrated Discriminative Classifiers](https://mlsanity.com/paper/1506.02169/) (2015)
- [Likelihood-free inference by ratio estimation](https://mlsanity.com/paper/1611.10242/) (2016)
- [Bayesian Optimization for Likelihood-Free Inference of Simulator-Based Statistical Models](https://mlsanity.com/paper/1501.03291/) (2015)
- [Likelihood-free inference with emulator networks](https://mlsanity.com/paper/1805.09294/) (2018)
- [Mining gold from implicit models to improve likelihood-free inference](https://mlsanity.com/paper/1805.12244/) (2018)
- [Universal Inference](https://mlsanity.com/paper/1912.11436/) (2019)
- [Diagnostics for Conditional Density Models and Bayesian Inference Algorithms](https://mlsanity.com/paper/2102.10473/) (2021)
- [Fast covariance parameter estimation of spatial Gaussian process models using neural networks](https://mlsanity.com/paper/2012.15339/) (2020)
- [High-Dimensional Density Ratio Estimation with Extensions to Approximate Likelihood Computation](https://mlsanity.com/paper/1404.7063/) (2014)
- [Parallel Gaussian process surrogate Bayesian inference with noisy likelihood evaluations](https://mlsanity.com/paper/1905.01252/) (2019)
- [Confidence Sets and Hypothesis Testing in a Likelihood-Free Inference Setting](https://mlsanity.com/paper/2002.10399/) (2020)
- [GPS-ABC: Gaussian Process Surrogate Approximate Bayesian Computation](https://mlsanity.com/paper/1401.2838/) (2014)
- [Frequentist coverage and sup-norm convergence rate in Gaussian process regression](https://mlsanity.com/paper/1708.04753/) (2017)
- [ABC random forests for Bayesian parameter inference](https://mlsanity.com/paper/1605.05537/) (2016)
- [ABC-CDE: Towards Approximate Bayesian Computation with Complex High-Dimensional Data and Limited Simulations](https://mlsanity.com/paper/1805.05480/) (2018)
- [Accelerating ABC methods using Gaussian processes](https://mlsanity.com/paper/1401.1436/) (2014)
- [Adaptive Gaussian Copula ABC](https://mlsanity.com/paper/1902.10704/) (2019)
- [An Extended Empirical Saddlepoint Approximation for Intractable Likelihoods](https://mlsanity.com/paper/1601.01849/) (2016)
- [Approximating posteriors with high-dimensional nuisance parameters via integrated rotated Gaussian approximation](https://mlsanity.com/paper/1909.06753/) (2019)

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

- [Neural Methods for Amortized Inference](https://mlsanity.com/paper/2404.12484/) (2024)
- [Epistemic Uncertainty in Conformal Scores: A Unified Approach](https://mlsanity.com/paper/2502.06995/) (2025)
- [Simulation-based Bayesian inference under model misspecification](https://mlsanity.com/paper/2503.12315/) (2025)
- [A Kernel-Based Conditional Two-Sample Test Using Nearest Neighbors (with Applications to Calibration, Regression Curves, and Simulation-Based Inference)](https://mlsanity.com/paper/2407.16550/) (2024)
- [CP4SBI: Local Conformal Calibration of Credible Sets in Simulation-Based Inference](https://mlsanity.com/paper/2508.17077/) (2025)
- [Conformal Calibration of Statistical Confidence Sets](https://mlsanity.com/paper/2411.19368/) (2024)
- [Trustworthy scientific inference with generative models](https://mlsanity.com/paper/2508.02602/) (2025)
- [An Introduction to Bayesian and Frequentist Simulation-Based Inference with Machine Learning](https://mlsanity.com/paper/2607.21702/) (2026)
- [Likelihood-free inference with nuisance parameters through normalizing flows](https://mlsanity.com/paper/2609.10534/) (2026)
- [A Three-Way Testing Framework for Quantifying Epistemic Calibration Uncertainty in SBI](https://mlsanity.com/paper/2609.24419/) (2026)
- [SLICE: Sequential Likelihood-Free Inference via Conformal E-Values](https://mlsanity.com/paper/2610.13855/) (2026)

## Abstract (arXiv)

Many areas of science rely on simulators that implicitly encode intractable likelihood functions of complex systems. Classical statistical methods are poorly suited for these so-called likelihood-free inference (LFI) settings, especially outside asymptotic and low-dimensional regimes. At the same time, popular LFI methods - such as Approximate Bayesian Computation or more recent machine learning techniques - do not necessarily lead to valid scientific inference because they do not guarantee confidence sets with nominal coverage in general settings. In addition, LFI currently lacks practical diagnostic tools to check the actual coverage of computed confidence sets across the entire parameter space. In this work, we propose a modular inference framework that bridges classical statistics and modern machine learning to provide (i) a practical approach for constructing confidence sets with near finite-sample validity at any value of the unknown parameters, and (ii) interpretable diagnostics for estimating empirical coverage across the entire parameter space. We refer to this framework as likelihood-free frequentist inference (LF2I). Any method that defines a test statistic can leverage LF2I to create valid confidence sets and diagnostics without costly Monte Carlo or bootstrap samples at fixed parameter settings. We study two likelihood-based test statistics (ACORE and BFF) and demonstrate their performance on high-dimensional complex data. Code is available at https://github.com/lee-group-cmu/lf2i.