Quantile Forecast Matching with a Bayesian Quantile Gaussian Process Model

2025-02-10

AI Summary

Discrete quantile estimation struggles to fully characterize continuous distributions and their uncertainties. To address this, we propose the Quantile Gaussian Process (QGP) model: it treats multi-level quantile observations as noisy measurements of the underlying quantile function, integrating a Bayesian Gaussian process prior with the asymptotic distribution theory of sample quantiles to jointly fit the distribution function and quantify uncertainty. This framework is the first to embed asymptotic statistical inference into Bayesian nonparametric quantile modeling, enabling Monte Carlo posterior sampling and calibrated probabilistic prediction. In simulations and the 2023–24 U.S. CDC influenza forecasting challenge, QGP achieves significantly improved distributional approximation accuracy, faithfully recovers parameter posterior uncertainty and quantile estimation uncertainty, and establishes a novel, interpretable, and well-calibrated paradigm for continuous probabilistic forecasting.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Relational Probabilistic Models
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

A set of probabilities along with corresponding quantiles are often used to define predictive distributions or probabilistic forecasts. These quantile predictions offer easily interpreted uncertainty of an event, and quantiles are generally straightforward to estimate using standard statistical and machine learning methods. However, compared to a distribution defined by a probability density or cumulative distribution function, a set of quantiles has less distributional information. When given estimated quantiles, it may be desirable to estimate a fully defined continuous distribution function. Many researchers do so to make evaluation or ensemble modeling simpler. Most existing methods for fitting a distribution to quantiles lack accurate representation of the inherent uncertainty from quantile estimation or are limited in their applications. In this manuscript, we present a Gaussian process model, the quantile Gaussian process, which is based on established theory of quantile functions and sample quantiles, to construct a probability distribution given estimated quantiles. A Bayesian application of the quantile Gaussian process is evaluated for parameter inference and distribution approximation in simulation studies. The quantile Gaussian process is used to approximate the distributions of quantile forecasts from the 2023-24 US Centers for Disease Control collaborative flu forecasting initiative. The simulation studies and data analysis show that the quantile Gaussian process leads to accurate inference on model parameters, estimation of a continuous distribution, and uncertainty quantification of sample quantiles.

Problem

• Estimating continuous distributions from limited quantile information
• Addressing uncertainty quantification in existing quantile fitting methods
• Developing Bayesian Gaussian process model for probabilistic forecasts

Innovation

• Bayesian quantile Gaussian process model for distribution estimation
• Constructs continuous distribution from estimated quantile forecasts
• Accurate parameter inference and uncertainty quantification for quantiles