# Generalized Bhattacharyya and Chernoff upper bounds on Bayes error using quasi-arithmetic means

arXiv: https://arxiv.org/abs/1401.4788 · HTML: https://mlsanity.com/paper/1401.4788/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jan 2014.

## Built on by (most cited first, 11 of 11)

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- [How Does Bayes Error Limit Probabilistic Robust Accuracy](https://mlsanity.com/paper/2405.14923/) (2024)
- [On Optimal Steering to Achieve Exact Fairness](https://mlsanity.com/paper/2509.15759/) (2025)
- [VERSE: Verified Self-Evolving Optimizer for Agent Harnesses](https://mlsanity.com/paper/2610.02616/) (2026)
- [Two tales for a geometric Jensen--Shannon divergence](https://mlsanity.com/paper/2508.05066/) (2025)
- [End-to-End Optimization of Incoherent Imaging for Classification Under Detector-Limited Readout](https://mlsanity.com/paper/2606.09792/) (2026)
- [When to Truncate a Feature Ranking: A Residual-Overlap Stopping Rule for Subset Selection](https://mlsanity.com/paper/2606.31686/) (2026)
- [The Ceiling Is in the Channel: Auditing Learner Gaps and Measurement Frontiers in Clinical Prediction](https://mlsanity.com/paper/2609.01909/) (2026)

## Abstract (arXiv)

Bayesian classification labels observations based on given prior information, namely class-a priori and class-conditional probabilities. Bayes' risk is the minimum expected classification cost that is achieved by the Bayes' test, the optimal decision rule. When no cost incurs for correct classification and unit cost is charged for misclassification, Bayes' test reduces to the maximum a posteriori decision rule, and Bayes risk simplifies to Bayes' error, the probability of error. Since calculating this probability of error is often intractable, several techniques have been devised to bound it with closed-form formula, introducing thereby measures of similarity and divergence between distributions like the Bhattacharyya coefficient and its associated Bhattacharyya distance. The Bhattacharyya upper bound can further be tightened using the Chernoff information that relies on the notion of best error exponent. In this paper, we first express Bayes' risk using the total variation distance on scaled distributions. We then elucidate and extend the Bhattacharyya and the Chernoff upper bound mechanisms using generalized weighted means. We provide as a byproduct novel notions of statistical divergences and affinity coefficients. We illustrate our technique by deriving new upper bounds for the univariate Cauchy and the multivariate t-distributions, and show experimentally that those bounds are not too distant to the computationally intractable Bayes' error.