Valid prediction intervals for regression problems (2107.00363v4)

Abstract: Over the last few decades, various methods have been proposed for estimating prediction intervals in regression settings, including Bayesian methods, ensemble methods, direct interval estimation methods and conformal prediction methods. An important issue is the calibration of these methods: the generated prediction intervals should have a predefined coverage level, without being overly conservative. In this work, we review the above four classes of methods from a conceptual and experimental point of view. Results on benchmark data sets from various domains highlight large fluctuations in performance from one data set to another. These observations can be attributed to the violation of certain assumptions that are inherent to some classes of methods. We illustrate how conformal prediction can be used as a general calibration procedure for methods that deliver poor results without a calibration step.

• The paper conducts a comparative study of Bayesian, ensemble, direct, and conformal prediction methods for estimating valid prediction intervals in regression tasks.
• Empirical analysis shows no single best method across all datasets, but conformal prediction robustly delivers valid intervals.
• Achieving valid intervals is crucial for reliability but faces challenges like violating model assumptions and scaling to complex data.

Valid Prediction Intervals for Regression Problems

The research paper titled "Valid prediction intervals for regression problems" by Nicolas Dewolf, Bernard De Baets, and Willem Waegeman presents an extensive analysis of various methods to estimate prediction intervals in the field of regression tasks. This comparative paper focuses on four common methodologies: Bayesian methods, ensemble methods, direct interval estimation methods, and conformal prediction methods. The authors emphasize the necessity for calibrated and valid prediction intervals, which provide a predefined coverage level without being overly conservative.

Overview and Methodology

The authors systematically evaluate the four classes of methods within an i.i.d. setting to assess their performance on diverse benchmark data sets. Despite significant advancements in uncertainty quantification, existing techniques typically deal with classification problems. However, the research pivots the focus to regression tasks where the complexity lies within adequately estimating prediction intervals instead of simple probability outputs.

The following methods were considered:

1. Bayesian Methods: These methods utilize the posterior predictive distribution from Bayesian inference to provide prediction intervals, offering robust theoretical guarantees when priors are accurately specified. However, real-world application often necessitates approximations due to computational infeasibility.
2. Ensemble Methods: By aggregating predictions from multiple models, ensemble methods like random forests and dropout networks provide better predictive performance. However, deriving valid uncertainty bounds from these ensembles is non-trivial, as standard deviation-based techniques might not fulfill the desired valid coverage properties without adjustments.
3. Direct Interval Estimation Methods: Methods in this category focus on directly estimating prediction intervals using loss functions like the pinball loss for quantile regression, ensuring intuitive coverage by design.
4. Conformal Prediction Methods: This framework applies a post-hoc calibration to existing predictions, ensuring valid prediction intervals empirically by leveraging nonconformity scores, thus making any point or interval predictor valid with respect to its coverage.

Empirical Assessment

The empirical analysis showcases substantial findings:

• There is no single method universally superior across all data sets, demonstrating the significance of data characteristics on model performance.
• Conformal prediction, used for calibrating other interval prediction methods, robustly delivers valid prediction intervals across diverse scenarios.
• Models incorporating probabilistic frameworks, such as Bayesian or deep ensembles, tend to better handle real-world data complexities, but still often require post hoc calibration to achieve desired validity.
• Challenges arise primarily due to model assumptions, such as distributional symmetry, which when violated can lead to invalid predictions.

Implications and Future Directions

The paper underscores the paramount importance of valid prediction intervals in regression, especially in applications necessitating high reliability, such as in safety-critical systems. While the research provides a detailed evaluation of various methodologies, it also prompts further investigation into scaling these techniques for larger, more complex data environments and relaxing strong assumptions, such as i.i.d., while preserving validity.

In future developments, exploring hybrid approaches that combine calibrated predictive intervals with advanced learning models might offer enhanced predictive reliability. Additionally, continuous research could aim to extend these methodologies to non-i.i.d. scenarios often encountered in time-series analyses, potentially broadening the scope of applications for valid prediction intervals in the field.