# The fundamental nature of the log loss function

arXiv: https://arxiv.org/abs/1502.06254 · HTML: https://mlsanity.com/paper/1502.06254/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Feb 2015.

## Built on by (most cited first, 10 of 10)

- [Neural Unification for Logic Reasoning over Natural Language](https://mlsanity.com/paper/2109.08460/) (2021)
- [FMFCC-A: A Challenging Mandarin Dataset for Synthetic Speech Detection](https://mlsanity.com/paper/2110.09441/) (2021)
- [CACTUS: a Comprehensive Abstraction and Classification Tool for Uncovering Structures](https://mlsanity.com/paper/2308.12031/) (2023)
- [Bayesian Classifier Fusion with an Explicit Model of Correlation](https://mlsanity.com/paper/2106.01770/) (2021)
- [Protected probabilistic classification](https://mlsanity.com/paper/2107.01726/) (2021)
- [T2CI-GAN: Text to Compressed Image generation using Generative Adversarial Network](https://mlsanity.com/paper/2210.03734/) (2022)
- [The Geometry of Mixability](https://mlsanity.com/paper/2302.11905/) (2023)
- [An Axiomatic Approach to Loss Aggregation and an Adapted Aggregating Algorithm](https://mlsanity.com/paper/2406.02292/) (2024)
- [Calibeating for general proper losses: A Bregman divergence approach](https://mlsanity.com/paper/2605.17269/) (2026)
- [A Hybrid Approach For Malware Classification Using Secondary Features Fusion](https://mlsanity.com/paper/2606.03432/) (2026)

## Abstract (arXiv)

The standard loss functions used in the literature on probabilistic prediction are the log loss function, the Brier loss function, and the spherical loss function; however, any computable proper loss function can be used for comparison of prediction algorithms. This note shows that the log loss function is most selective in that any prediction algorithm that is optimal for a given data sequence (in the sense of the algorithmic theory of randomness) under the log loss function will be optimal under any computable proper mixable loss function; on the other hand, there is a data sequence and a prediction algorithm that is optimal for that sequence under either of the two other standard loss functions but not under the log loss function.