# Interval-based Prediction Uncertainty Bound Computation in Learning with Missing Values

## TL;DR

This paper introduces the IPUB method, which efficiently computes bounds on prediction uncertainty in machine learning models with missing data by representing uncertainties as intervals, improving over naive and multiple imputation approaches.

## Contribution

The paper proposes the IPUB method that provides interval-based bounds on prediction uncertainty, applicable to various convex learning algorithms, offering a computationally efficient alternative to existing methods.

## Key findings

- IPUB effectively bounds prediction uncertainty in missing data scenarios.
- Compared to existing methods, IPUB shows improved efficiency and comparable accuracy.
- Numerical experiments demonstrate the advantages of IPUB on benchmark datasets.

## Abstract

The problem of machine learning with missing values is common in many areas. A simple approach is to first construct a dataset without missing values simply by discarding instances with missing entries or by imputing a fixed value for each missing entry, and then train a prediction model with the new dataset. A drawback of this naive approach is that the uncertainty in the missing entries is not properly incorporated in the prediction. In order to evaluate prediction uncertainty, the multiple imputation (MI) approach has been studied, but the performance of MI is sensitive to the choice of the probabilistic model of the true values in the missing entries, and the computational cost of MI is high because multiple models must be trained. In this paper, we propose an alternative approach called the Interval-based Prediction Uncertainty Bounding (IPUB) method. The IPUB method represents the uncertainties due to missing entries as intervals, and efficiently computes the lower and upper bounds of the prediction results when all possible training sets constructed by imputing arbitrary values in the intervals are considered. The IPUB method can be applied to a wide class of convex learning algorithms including penalized least-squares regression, support vector machine (SVM), and logistic regression. We demonstrate the advantages of the IPUB method by comparing it with an existing method in numerical experiment with benchmark datasets.