# Calibrated Multi-Level Quantile Forecasting

arXiv: https://arxiv.org/abs/2512.23671 · HTML: https://mlsanity.com/paper/2512.23671/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 54 references that are in the map.
- First posted on arXiv 29 Dec 2025.

## Builds on (most cited first, 24 of 24)

- [Introduction to Online Convex Optimization](https://mlsanity.com/paper/1909.05207/) (2019)
- [Adaptive Conformal Inference Under Distribution Shift](https://mlsanity.com/paper/2106.00170/) (2021)
- [Adaptive Conformal Predictions for Time Series](https://mlsanity.com/paper/2202.07282/) (2022)
- [Conformal Inference for Online Prediction with Arbitrary Distribution Shifts](https://mlsanity.com/paper/2208.08401/) (2022)
- [Conformal PID Control for Time Series Prediction](https://mlsanity.com/paper/2307.16895/) (2023)
- [Improved Online Conformal Prediction via Strongly Adaptive Online Learning](https://mlsanity.com/paper/2302.07869/) (2023)
- [Practical Adversarial Multivalid Conformal Prediction](https://mlsanity.com/paper/2206.01067/) (2022)
- [Conformal Decision Theory: Safe Autonomous Decisions from Imperfect Predictions](https://mlsanity.com/paper/2310.05921/) (2023)
- [Online Multivalid Learning: Means, Moments, and Prediction Intervals](https://mlsanity.com/paper/2101.01739/) (2021)
- [High-Dimensional Prediction for Sequential Decision Making](https://mlsanity.com/paper/2310.17651/) (2023)
- [Achieving Risk Control in Online Learning Settings](https://mlsanity.com/paper/2205.09095/) (2022)
- [Learning Quantile Functions without Quantile Crossing for Distribution-free Time Series Forecasting](https://mlsanity.com/paper/2111.06581/) (2021)
- [Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination](https://mlsanity.com/paper/2108.03210/) (2021)
- [Flexible Model Aggregation for Quantile Regression](https://mlsanity.com/paper/2103.00083/) (2021)
- [Forecast Hedging and Calibration](https://mlsanity.com/paper/2210.07169/) (2022)
- [Bellman Conformal Inference: Calibrating Prediction Intervals For Time Series](https://mlsanity.com/paper/2402.05203/) (2024)
- [Online conformal inference for multi-step time series forecasting](https://mlsanity.com/paper/2410.13115/) (2024)
- [Multi-model Ensemble Conformal Prediction in Dynamic Environments](https://mlsanity.com/paper/2411.03678/) (2024)
- [Calibrated Probabilistic Forecasts for Arbitrary Sequences](https://mlsanity.com/paper/2409.19157/) (2024)
- [Neural Conformal Control for Time Series Forecasting](https://mlsanity.com/paper/2412.18144/) (2024)
- [Calibrated Regression Against An Adversary Without Regret](https://mlsanity.com/paper/2302.12196/) (2023)
- [The Benefit of Being Bayesian in Online Conformal Prediction](https://mlsanity.com/paper/2410.02561/) (2024)
- [Recalibrating probabilistic forecasts of epidemics](https://mlsanity.com/paper/2112.06305/) (2021)
- [Distribution-informed Online Conformal Prediction](https://mlsanity.com/paper/2512.07770/) (2025)

## Built on by (most cited first, 2 of 2)

- [Risk-Averse Decision Making with Multi-Level Reliability Guarantees](https://mlsanity.com/paper/2609.11524/) (2026)
- [Blackwell Approachability and Gradient Equilibrium are Equivalent](https://mlsanity.com/paper/2606.27315/) (2026)

## Abstract (arXiv)

We develop an online method that guarantees calibration of quantile forecasts at multiple quantile levels simultaneously. In this work, a sequence of quantile forecasts is said to be calibrated provided that its α-level predictions are greater than or equal to the target value at an α fraction of time steps, for each level α. Our procedure, called the multi-level quantile tracker (MultiQT), is lightweight and wraps around any point or quantile forecaster to produce adjusted quantile forecasts that are guaranteed to be calibrated, even against adversarial distribution shifts. Critically, it does so while ensuring that the quantiles remain ordered, e.g., the 0.5-level quantile forecast will never be larger than the 0.6-level forecast. Moreover, the method has a no-regret guarantee, implying it will not degrade the performance of the existing forecaster (asymptotically), with respect to the quantile loss. In our experiments, we find that MultiQT significantly improves the calibration of real forecasters in epidemic and energy forecasting problems, while leaving the quantile loss largely unchanged or slightly improved.