# Distribution-free inference with hierarchical data

arXiv: https://arxiv.org/abs/2306.06342 · HTML: https://mlsanity.com/paper/2306.06342/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2023.

## Built on by (most cited first, 15 of 15)

- [Kernel-based Optimally Weighted Conformal Time-Series Prediction](https://mlsanity.com/paper/2405.16828/) (2024)
- [Multi-Source Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2405.09331/) (2024)
- [Conformal Structured Prediction](https://mlsanity.com/paper/2410.06296/) (2024)
- [Provable Uncertainty Decomposition via Higher-Order Calibration](https://mlsanity.com/paper/2412.18808/) (2024)
- [SymmPI: Predictive Inference for Data with Group Symmetries](https://mlsanity.com/paper/2312.16160/) (2023)
- [Conformal Prediction for Hierarchical Data](https://mlsanity.com/paper/2411.13479/) (2024)
- [Predictive Inference in Multi-environment Scenarios](https://mlsanity.com/paper/2403.16336/) (2024)
- [Uncertainty Quantification With Multiple Sources](https://mlsanity.com/paper/2405.06479/) (2024)
- [Conditional Predictive Inference for General Structured Data with Group Symmetries](https://mlsanity.com/paper/2605.17934/) (2026)
- [Conformalized Quantum DeepONet Ensembles: Towards Scalable Operator Learning with Distribution-Free Guarantees](https://mlsanity.com/paper/2605.00330/) (2026)
- [Full Conformal Prediction under Stochastic Non-Conformity Measure](https://mlsanity.com/paper/2606.28730/) (2026)
- [Generalized Hierarchical Conformal Prediction](https://mlsanity.com/paper/2608.15500/) (2026)
- [The Exceedance Design Effect: Effective Sample Size for Thresholds under Clustering](https://mlsanity.com/paper/2608.21262/) (2026)
- [Multi-source conformal prediction: leveraging heterogeneity via localization](https://mlsanity.com/paper/2609.14531/) (2026)
- [Missingness-Aware Conformal Prediction Under Cross-Hospital Distribution Shift](https://mlsanity.com/paper/2609.30781/) (2026)

## Abstract (arXiv)

This paper studies distribution-free inference in settings where the data set has a hierarchical structure -- for example, groups of observations, or repeated measurements. In such settings, standard notions of exchangeability may not hold. To address this challenge, a hierarchical form of exchangeability is derived, facilitating extensions of distribution-free methods, including conformal prediction and jackknife+. While the standard theoretical guarantee obtained by the conformal prediction framework is a marginal predictive coverage guarantee, in the special case of independent repeated measurements, it is possible to achieve a stronger form of coverage -- the "second-moment coverage" property -- to provide better control of conditional miscoverage rates, and distribution-free prediction sets that achieve this property are constructed. Simulations illustrate that this guarantee indeed leads to uniformly small conditional miscoverage rates. Empirically, this stronger guarantee comes at the cost of a larger width of the prediction set in scenarios where the fitted model is poorly calibrated, but this cost is very mild in cases where the fitted model is accurate.