Distribution-Free Predictive Inference For Regression (1604.04173v2)

Abstract: We develop a general framework for distribution-free predictive inference in regression, using conformal inference. The proposed methodology allows for the construction of a prediction band for the response variable using any estimator of the regression function. The resulting prediction band preserves the consistency properties of the original estimator under standard assumptions, while guaranteeing finite-sample marginal coverage even when these assumptions do not hold. We analyze and compare, both empirically and theoretically, the two major variants of our conformal framework: full conformal inference and split conformal inference, along with a related jackknife method. These methods offer different tradeoffs between statistical accuracy (length of resulting prediction intervals) and computational efficiency. As extensions, we develop a method for constructing valid in-sample prediction intervals called {\it rank-one-out} conformal inference, which has essentially the same computational efficiency as split conformal inference. We also describe an extension of our procedures for producing prediction bands with locally varying length, in order to adapt to heteroskedascity in the data. Finally, we propose a model-free notion of variable importance, called {\it leave-one-covariate-out} or LOCO inference. Accompanying this paper is an R package {\tt conformalInference} that implements all of the proposals we have introduced. In the spirit of reproducibility, all of our empirical results can also be easily (re)generated using this package.

Distribution-Free Predictive Inference For Regression

The paper "Distribution-Free Predictive Inference For Regression," authored by Jing Lei, Max G'Sell, Alessandro Rinaldo, Ryan J. Tibshirani, and Larry Wasserman from Carnegie Mellon University, presents a framework for predictive inference in regression using conformal inference. The key contribution is the development of finite-sample valid prediction bands that do not rely on distributional assumptions about the underlying data generating process.

Key Contributions and Methodology

1. Framework for Distribution-Free Predictive Inference: The authors introduce a conformal predictive inference approach that utilizes any regression function estimator to construct prediction intervals. This methodology ensures finite-sample marginal coverage without assumptions on the data distribution or the regression model, preserving the estimator's consistency under standard assumptions.
2. Variants of Conformal Inference: The paper empirically and theoretically analyzes two primary variants: full conformal inference and split conformal inference. The full conformal method involves retraining the model for each new prediction, ensuring accurate coverage but at a high computational cost. Split conformal inference, on the other hand, separates the sample into two parts, training on one and validating on the other, offering a significant reduction in computational expense while still maintaining the desired coverage.
3. Extensions to Conformal Inference: The authors propose methods for constructing valid in-sample prediction intervals and prediction intervals with locally varying lengths to account for heteroskedasticity. The proposed rank-one-out (ROO) conformal inference is almost as efficient as split conformal inference and allows for flexible adaptation to changing data variances.
4. Model-Free Variable Importance (LOCO): The paper introduces a new notion of variable importance called leave-one-covariate-out (LOCO) inference. This approach assesses the marginal contribution of each covariate to the model's predictive performance by comparing prediction errors with and without the covariate in question. LOCO provides distribution-free, finite-sample valid intervals for variable importance.

Numerical Results and Empirical Comparisons

The empirical results demonstrate that conformal predictive intervals maintain the desired coverage across various settings, including high-dimensional regression scenarios where traditional methods fall short due to model misspecification or computational infeasibility. The paper includes simulations comparing the length and coverage of conformal intervals to classical parametric intervals in both low- and high-dimensional settings. In high-dimensional settings, conformal methods outperform classical methods, offering valid inference without making stringent assumptions.

Theoretical Contributions

The paper offers several theoretical results to substantiate the empirical findings:

• Finite-Sample Validity: Both full and split conformal methods are shown to provide valid coverage for prediction intervals, confirmed by the theoretical bounds presented.
• Accuracy Analysis: The authors provide near-optimal bounds on the length of prediction intervals under standard assumptions and demonstrate that conformal methods approximate certain oracle methods. Specifically, stability and consistency properties of the base estimator significantly influence the accuracy of prediction intervals.
• Multiple Splits: It is shown that multiple splits may increase interval lengths due to a Bonferroni-type correction effect, suggesting the use of a single split for efficiency.

Practical and Theoretical Implications

This work has significant implications for both theoretical and practical aspects of statistical learning and predictive modeling in high-dimensional spaces:

• Robust Predictive Inference: The proposed conformal inference framework ensures robust, finite-sample predictive inferences under minimal assumptions, making it particularly valuable in high-dimensional settings where model assumptions are typically violated.
• Efficient Computations: The split conformal method provides a powerful trade-off between computational efficiency and statistical accuracy, enabling its application in real-world high-dimensional datasets.
• Variable Importance: LOCO inference extends the framework to assess variable importance in a model-free manner, facilitating the evaluation of feature contributions in complex models.

Future Directions

The paper opens several avenues for future research, emphasizing the development of more efficient combination strategies for multiple splits, further investigations into model-free variable selection, and comprehensive comparisons between LOCO and other high-dimensional inference approaches. These advancements have the potential to extend the applicability of conformal inference methods to an even broader range of predictive modeling tasks.

This work stands out for its rigorous theoretical foundations coupled with practical applicability, providing a robust and flexible framework for predictive inference in high-dimensional statistical learning.