# Multivariate Confidence Intervals

## TL;DR

This paper introduces a natural extension of confidence intervals to multivariate data, allowing for more informative visualization and analysis of complex data distributions, with efficient algorithms for their computation.

## Contribution

It defines multivariate confidence intervals that generalize the univariate case, maintaining interpretability and providing approximate algorithms for their computation.

## Key findings

- Multivariate confidence areas are informative and easy to interpret.
- The problem of computing these intervals is computationally hard.
- Efficient approximate algorithms are proposed for practical use.

## Abstract

Confidence intervals are a popular way to visualize and analyze data distributions. Unlike p-values, they can convey information both about statistical significance as well as effect size. However, very little work exists on applying confidence intervals to multivariate data. In this paper we define confidence intervals for multivariate data that extend the one-dimensional definition in a natural way. In our definition every variable is associated with its own confidence interval as usual, but a data vector can be outside of a few of these, and still be considered to be within the confidence area. We analyze the problem and show that the resulting confidence areas retain the good qualities of their one-dimensional counterparts: they are informative and easy to interpret. Furthermore, we show that the problem of finding multivariate confidence intervals is hard, but provide efficient approximate algorithms to solve the problem.