# Multivariate Chebyshev Inequality with Estimated Mean and Variance

arXiv: https://arxiv.org/abs/1509.08398 · HTML: https://mlsanity.com/paper/1509.08398/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Sep 2015.

## Built on by (most cited first, 5 of 5)

- [Uncertainty Quantification in Scientific Machine Learning: Methods, Metrics, and Comparisons](https://mlsanity.com/paper/2201.07766/) (2022)
- [Non-Asymptotic Uncertainty Quantification in High-Dimensional Learning](https://mlsanity.com/paper/2407.13666/) (2024)
- [PICProp: Physics-Informed Confidence Propagation for Uncertainty Quantification](https://mlsanity.com/paper/2310.06923/) (2023)
- [Introducing the structural bases of typicality effects in deep learning](https://mlsanity.com/paper/2107.03279/) (2021)
- [TESDA: Transform Enabled Statistical Detection of Attacks in Deep Neural Networks](https://mlsanity.com/paper/2110.08447/) (2021)

## Abstract (arXiv)

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple dimensions where the only requirement is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.