# Testing Goodness of Fit of Conditional Density Models with Kernels

arXiv: https://arxiv.org/abs/2002.10271 · HTML: https://mlsanity.com/paper/2002.10271/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 24 Feb 2020.

## Built on by (most cited first, 19 of 19)

- [KSD Aggregated Goodness-of-fit Test](https://mlsanity.com/paper/2202.00824/) (2022)
- [Composite Goodness-of-fit Tests with Kernels](https://mlsanity.com/paper/2111.10275/) (2021)
- [Kernel Two-Sample Tests for Manifold Data](https://mlsanity.com/paper/2105.03425/) (2021)
- [Interpretable Uncertainty Quantification in AI for HEP](https://mlsanity.com/paper/2208.03284/) (2022)
- [Standardisation-function Kernel Stein Discrepancy: A Unifying View on Kernel Stein Discrepancy Tests for Goodness-of-fit](https://mlsanity.com/paper/2106.12105/) (2021)
- [STEERING: Stein Information Directed Exploration for Model-Based Reinforcement Learning](https://mlsanity.com/paper/2301.12038/) (2023)
- [Nystr"om Kernel Stein Discrepancy](https://mlsanity.com/paper/2406.08401/) (2024)
- [Re-calibrating Photometric Redshift Probability Distributions Using Feature-space Regression](https://mlsanity.com/paper/2110.15209/) (2021)
- [Data-centric Prediction Explanation via Kernelized Stein Discrepancy](https://mlsanity.com/paper/2403.15576/) (2024)
- [A Stein Goodness of fit Test for Exponential Random Graph Models](https://mlsanity.com/paper/2103.00580/) (2021)
- [On Parameter Estimation in Deviated Gaussian Mixture of Experts](https://mlsanity.com/paper/2402.05220/) (2024)
- [Score-based Generative Modeling for Conditional Independence Testing](https://mlsanity.com/paper/2505.23309/) (2025)
- [Fast and Scalable Score-Based Kernel Calibration Tests](https://mlsanity.com/paper/2510.14711/) (2025)
- [Kernel-Based Evaluation of Conditional Biological Sequence Models](https://mlsanity.com/paper/2510.15601/) (2025)
- [A moment-matching metric for latent variable generative models](https://mlsanity.com/paper/2111.00875/) (2021)
- [RFFNet: Large-Scale Interpretable Kernel Methods via Random Fourier Features](https://mlsanity.com/paper/2211.06410/) (2022)
- [Empirical Bayes Estimation with Side Information: A Nonparametric Integrative Tweedie Approach](https://mlsanity.com/paper/2308.05883/) (2023)
- [Correcting Mode Proportion Bias in Generalized Bayesian Inference via a Weighted Kernel Stein Discrepancy](https://mlsanity.com/paper/2503.02108/) (2025)
- [Kernel Quantile Embeddings and Associated Probability Metrics](https://mlsanity.com/paper/2505.20433/) (2025)

## Abstract (arXiv)

We propose two nonparametric statistical tests of goodness of fit for conditional distributions: given a conditional probability density function p(y|x) and a joint sample, decide whether the sample is drawn from p(y|x)rx(x) for some density rx. Our tests, formulated with a Stein operator, can be applied to any differentiable conditional density model, and require no knowledge of the normalizing constant. We show that 1) our tests are consistent against any fixed alternative conditional model; 2) the statistics can be estimated easily, requiring no density estimation as an intermediate step; and 3) our second test offers an interpretable test result providing insight on where the conditional model does not fit well in the domain of the covariate. We demonstrate the interpretability of our test on a task of modeling the distribution of New York City's taxi drop-off location given a pick-up point. To our knowledge, our work is the first to propose such conditional goodness-of-fit tests that simultaneously have all these desirable properties.