# Approximately Unimodal Likelihood Models for Ordinal Regression

arXiv: https://arxiv.org/abs/2510.00122 · HTML: https://mlsanity.com/paper/2510.00122/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 45 references that are in the map.
- First posted on arXiv 30 Sep 2025.

## Builds on (most cited first, 8 of 8)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Deep Ordinal Regression Network for Monocular Depth Estimation](https://mlsanity.com/paper/1806.02446/) (2018)
- [MedMNIST v2 -- A large-scale lightweight benchmark for 2D and 3D biomedical image classification](https://mlsanity.com/paper/2110.14795/) (2021)
- [Rank consistent ordinal regression for neural networks with application to age estimation](https://mlsanity.com/paper/1901.07884/) (2019)
- [Unimodal probability distributions for deep ordinal classification](https://mlsanity.com/paper/1705.05278/) (2017)
- [Parallel Algorithm for Optimal Threshold Labeling of Ordinal Regression Methods](https://mlsanity.com/paper/2405.12756/) (2024)
- [Remarks on Loss Function of Threshold Method for Ordinal Regression Problem](https://mlsanity.com/paper/2405.13288/) (2024)

## Built on by (most cited first, 1 of 1)

- [Unimodality-Promoting Regularized Learning for Ordinal Regression](https://mlsanity.com/paper/2608.08359/) (2026)

## Abstract (arXiv)

Ordinal regression (OR, also called ordinal classification) is classification of ordinal data, in which the underlying target variable is categorical and considered to have a natural ordinal relation for the underlying explanatory variable. A key to successful OR models is to find a data structure `natural ordinal relation' common to many ordinal data and reflect that structure into the design of those models. A recent OR study found that many real-world ordinal data show a tendency that the conditional probability distribution (CPD) of the target variable given a value of the explanatory variable will often be unimodal. Several previous studies thus developed unimodal likelihood models, in which a predicted CPD is guaranteed to become unimodal. However, it was also observed experimentally that many real-world ordinal data partly have values of the explanatory variable where the underlying CPD will be non-unimodal, and hence unimodal likelihood models may suffer from a bias for such a CPD. Therefore, motivated to mitigate such a bias, we propose approximately unimodal likelihood models, which can represent up to a unimodal CPD and a CPD that is close to be unimodal. We also verify experimentally that a proposed model can be effective for statistical modeling of ordinal data and OR tasks.