# Simulation-Based Inference with Quantile Regression

arXiv: https://arxiv.org/abs/2401.02413 · HTML: https://mlsanity.com/paper/2401.02413/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 54 references that are in the map.
- First posted on arXiv 4 Jan 2024.

## Builds on (most cited first, 28 of 28)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [FFJORD: Free-form Continuous Dynamics for Scalable Reversible Generative Models](https://mlsanity.com/paper/1810.01367/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Revisiting Classifier Two-Sample Tests](https://mlsanity.com/paper/1610.06545/) (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)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [Benchmarking Simulation-Based Inference](https://mlsanity.com/paper/2101.04653/) (2021)
- [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)
- [Flow Matching for Scalable Simulation-Based Inference](https://mlsanity.com/paper/2305.17161/) (2023)
- [On Contrastive Learning for Likelihood-free Inference](https://mlsanity.com/paper/2002.03712/) (2020)
- [Likelihood-free inference with emulator networks](https://mlsanity.com/paper/1805.09294/) (2018)
- [Learning Summary Statistic for Approximate Bayesian Computation via Deep Neural Network](https://mlsanity.com/paper/1510.02175/) (2015)
- [Beyond expectation: Deep joint mean and quantile regression for spatio-temporal problems](https://mlsanity.com/paper/1808.08798/) (2018)
- [Truncated Marginal Neural Ratio Estimation](https://mlsanity.com/paper/2107.01214/) (2021)
- [A Trust Crisis In Simulation-Based Inference? Your Posterior Approximations Can Be Unfaithful](https://mlsanity.com/paper/2110.06581/) (2021)
- [Towards Reliable Simulation-Based Inference with Balanced Neural Ratio Estimation](https://mlsanity.com/paper/2208.13624/) (2022)
- [Graphical Test for Discrete Uniformity and its Applications in Goodness of Fit Evaluation and Multiple Sample Comparison](https://mlsanity.com/paper/2103.10522/) (2021)
- [Solving high-dimensional parameter inference: marginal posterior densities & Moment Networks](https://mlsanity.com/paper/2011.05991/) (2020)
- [Adaptive Approximate Bayesian Computation Tolerance Selection](https://mlsanity.com/paper/1907.01505/) (2019)
- [Nonparametric Quantile Regression: Non-Crossing Constraints and Conformal Prediction](https://mlsanity.com/paper/2210.10161/) (2022)
- [Neural Spline Search for Quantile Probabilistic Modeling](https://mlsanity.com/paper/2301.04857/) (2023)

## Built on by (most cited first, 4 of 4)

- [Effortless, Simulation-Efficient Bayesian Inference using Tabular Foundation Models](https://mlsanity.com/paper/2504.17660/) (2025)
- [Simulation-based Inference with the Python Package sbijax](https://mlsanity.com/paper/2409.19435/) (2024)
- [Cosmological Analysis with Calibrated Neural Quantile Estimation and Approximate Simulators](https://mlsanity.com/paper/2411.14748/) (2024)
- [Simulation-Efficient Cosmological Inference with Multi-Fidelity SBI](https://mlsanity.com/paper/2507.00514/) (2025)

## Abstract (arXiv)

We present Neural Quantile Estimation (NQE), a novel Simulation-Based Inference (SBI) method based on conditional quantile regression. NQE autoregressively learns individual one dimensional quantiles for each posterior dimension, conditioned on the data and previous posterior dimensions. Posterior samples are obtained by interpolating the predicted quantiles using monotonic cubic Hermite spline, with specific treatment for the tail behavior and multi-modal distributions. We introduce an alternative definition for the Bayesian credible region using the local Cumulative Density Function (CDF), offering substantially faster evaluation than the traditional Highest Posterior Density Region (HPDR). In case of limited simulation budget and/or known model misspecification, a post-processing calibration step can be integrated into NQE to ensure the unbiasedness of the posterior estimation with negligible additional computational cost. We demonstrate that NQE achieves state-of-the-art performance on a variety of benchmark problems.