conformal inference confidence interval of quantile

Lemon

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03-02-2025

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Conformal inference provides a powerful and versatile framework for constructing prediction intervals and confidence intervals, even for complex models where traditional methods struggle. This post delves into how conformal inference can be used to generate confidence intervals specifically for quantiles, offering a robust and adaptable approach to uncertainty quantification.

What is Conformal Inference?

Conformal inference is a non-parametric method for constructing prediction regions that possess a guaranteed coverage probability. Unlike traditional methods that rely on strong distributional assumptions, conformal inference makes minimal assumptions about the underlying data-generating process. This robustness makes it particularly appealing for complex models or situations with limited data. The core idea is to calibrate the predictions of any model (even a black box) using a conformity measure, which assesses how well a new data point conforms to the training data.

Quantile Estimation and Conformal Inference

Estimating quantiles accurately is crucial in many applications, from risk assessment to financial modeling. Traditional methods often rely on asymptotic normality assumptions, which can be violated in practice, especially with smaller datasets or complex data distributions. Conformal inference offers a valuable alternative.

Here's how it works for quantile estimation:

1. Training Set: We start with a training dataset used to train a quantile regression model (or any other suitable model capable of quantile prediction).

2. Conformity Score: For each data point in the training set, we calculate a conformity score. This score measures how "conformal" the point is to the rest of the data. Common conformity scores include the absolute prediction error or a more sophisticated measure based on the model's residuals.

3. Calibration Set: A separate calibration set is used to calibrate the conformity scores. This helps to determine the appropriate threshold for the conformity score that guarantees the desired coverage probability for the confidence interval.

4. Confidence Interval: For a new data point, the model predicts the quantile. The confidence interval is then constructed by considering all points in the calibration set with a conformity score below a certain threshold determined by the desired coverage probability (e.g., 95%). This effectively creates a prediction interval that contains the true quantile with a guaranteed probability.

Advantages of Conformal Inference for Quantile Confidence Intervals

• Non-parametric: It doesn't rely on strong distributional assumptions about the data. This makes it robust to model misspecification and deviations from normality.

• Versatile: It can be applied to any prediction model, even black box models where the internal workings are not fully understood.

• Guaranteed Coverage: The method guarantees the desired coverage probability of the confidence interval, regardless of the underlying data distribution.

• Adaptability: The choice of the conformity score can be tailored to specific applications and data characteristics.

Limitations of Conformal Inference

• Computational Cost: Calculating conformity scores for a large calibration set can be computationally expensive.

• Calibration Set Size: The size of the calibration set impacts the accuracy of the confidence interval. A larger calibration set generally leads to more reliable results but requires more data.

• Choice of Conformity Score: The selection of an appropriate conformity score can influence the performance of the method, requiring careful consideration of the specific application.

Conclusion

Conformal inference presents a robust and versatile approach to constructing confidence intervals for quantiles, overcoming limitations associated with traditional methods that rely on restrictive assumptions. Its non-parametric nature and guaranteed coverage probability make it particularly attractive in situations involving complex models or uncertain data distributions. While computational considerations and the choice of conformity score require attention, the benefits of its robustness and adaptability make conformal inference a valuable tool in the statistical arsenal. Further research is ongoing to optimize its efficiency and explore its applications across various domains.