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Guaranteed Coverage Prediction Intervals with Gaussian Process Regression

Harris Papadopoulos

TL;DR

This work tackles the problem of miscalibrated uncertainty in Gaussian Process Regression under model misspecification. It introduces GPR-CP, a transductive conformal predictor that uses a normalized nonconformity measure based on predictive variance to guarantee finite-sample coverage at level $1-\\delta$ under exchangeability, while preserving GPR's strengths. Empirical results on synthetic data and UCI benchmarks across several covariance functions show well-calibrated prediction intervals with competitive, often tighter widths compared to standard GPR and existing conformal or recalibration methods. The approach delivers robust uncertainty quantification for regression in settings with limited or misspecified models and points to future work on sparse GP implementations and further normalization schemes.

Abstract

Gaussian Process Regression (GPR) is a popular regression method, which unlike most Machine Learning techniques, provides estimates of uncertainty for its predictions. These uncertainty estimates however, are based on the assumption that the model is well-specified, an assumption that is violated in most practical applications, since the required knowledge is rarely available. As a result, the produced uncertainty estimates can become very misleading; for example the prediction intervals (PIs) produced for the 95% confidence level may cover much less than 95% of the true labels. To address this issue, this paper introduces an extension of GPR based on a Machine Learning framework called, Conformal Prediction (CP). This extension guarantees the production of PIs with the required coverage even when the model is completely misspecified. The proposed approach combines the advantages of GPR with the valid coverage guarantee of CP, while the performed experimental results demonstrate its superiority over existing methods.

Guaranteed Coverage Prediction Intervals with Gaussian Process Regression

TL;DR

This work tackles the problem of miscalibrated uncertainty in Gaussian Process Regression under model misspecification. It introduces GPR-CP, a transductive conformal predictor that uses a normalized nonconformity measure based on predictive variance to guarantee finite-sample coverage at level under exchangeability, while preserving GPR's strengths. Empirical results on synthetic data and UCI benchmarks across several covariance functions show well-calibrated prediction intervals with competitive, often tighter widths compared to standard GPR and existing conformal or recalibration methods. The approach delivers robust uncertainty quantification for regression in settings with limited or misspecified models and points to future work on sparse GP implementations and further normalization schemes.

Abstract

Gaussian Process Regression (GPR) is a popular regression method, which unlike most Machine Learning techniques, provides estimates of uncertainty for its predictions. These uncertainty estimates however, are based on the assumption that the model is well-specified, an assumption that is violated in most practical applications, since the required knowledge is rarely available. As a result, the produced uncertainty estimates can become very misleading; for example the prediction intervals (PIs) produced for the 95% confidence level may cover much less than 95% of the true labels. To address this issue, this paper introduces an extension of GPR based on a Machine Learning framework called, Conformal Prediction (CP). This extension guarantees the production of PIs with the required coverage even when the model is completely misspecified. The proposed approach combines the advantages of GPR with the valid coverage guarantee of CP, while the performed experimental results demonstrate its superiority over existing methods.
Paper Structure (11 sections, 1 theorem, 24 equations, 5 figures, 4 tables, 1 algorithm)

This paper contains 11 sections, 1 theorem, 24 equations, 5 figures, 4 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Conformal Prediction
  4. Related Work
  5. Gaussian Process Regression CP
  6. A Normalized Nonconformity Measure
  7. Experimental Evaluation
  8. Evaluation Metrics
  9. Artificial Data
  10. Benchmark Data
  11. Conclusion

Key Result

Theorem 1

(Conformal Prediction coverage guarantee vovk:alrw). Suppose $(x_1, y_1), \dots, (x_{l+1}, y_{l+1})$ are exchangeable, then the Conformal Prediction region (eq:predregion) satisfies

Figures (5)

  • Figure 1: Correspondence of $N$ and $M$ to the intervals and points of $P$.
  • Figure 2: Boston Housing data set PI width distribution for the $90\%$, $95\%$ and $99\%$ confidence levels (Each part contains the boxplots for the PI widths produced with $\gamma$ set to $1$, $2$, $3$, $4$, $8$ and $\infty$ from left to right).
  • Figure 3: Auto-mpg data set PI width distribution for the $90\%$, $95\%$ and $99\%$ confidence levels (Each part contains the boxplots for the PI widths produced with $\gamma$ set to $1$, $2$, $3$, $4$, $8$ and $\infty$ from left to right).
  • Figure 4: CPU Performance data set PI width distribution for the $90\%$, $95\%$ and $99\%$ confidence levels (Each part contains the boxplots for the PI widths produced with $\gamma$ set to $1$, $2$, $3$, $4$, $8$ and $\infty$ from left to right).
  • Figure 5: Servo data set PI width distribution for the $90\%$, $95\%$ and $99\%$ confidence levels (Each part contains the boxplots for the PI widths produced with $\gamma$ set to $1$, $2$, $3$, $4$, $8$ and $\infty$ from left to right).

Theorems & Definitions (2)

  • Theorem 1
  • Remark 1