# Learning Mixtures of Plackett-Luce Models from Structured Partial Orders

arXiv: https://arxiv.org/abs/1910.11721 · HTML: https://mlsanity.com/paper/1910.11721/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Oct 2019.

## Built on by (most cited first, 9 of 9)

- [Distortion of AI Alignment: Does Preference Optimization Optimize for Preferences?](https://mlsanity.com/paper/2505.23749/) (2025)
- [Learning Linear Utility Functions From Pairwise Comparison Queries](https://mlsanity.com/paper/2405.02612/) (2024)
- [Statistical Depth Functions for Ranking Distributions: Definitions, Statistical Learning and Applications](https://mlsanity.com/paper/2201.08105/) (2022)
- [Adaptive Sampling for Heterogeneous Rank Aggregation from Noisy Pairwise Comparisons](https://mlsanity.com/paper/2110.04136/) (2021)
- [Efficient and Accurate Learning of Mixtures of Plackett-Luce Models](https://mlsanity.com/paper/2302.05343/) (2023)
- [Statistical Models of Top-k Partial Orders](https://mlsanity.com/paper/2406.15893/) (2024)
- [When the Universe is Too Big: Bounding Consideration Probabilities for Plackett-Luce Rankings](https://mlsanity.com/paper/2401.11016/) (2024)
- [PLR: Plackett-Luce for Reordering In-Context Learning Examples](https://mlsanity.com/paper/2603.21373/) (2026)
- [Efficient and Accurate Top-K Recovery from Choice Data](https://mlsanity.com/paper/2206.11995/) (2022)

## Abstract (arXiv)

Mixtures of ranking models have been widely used for heterogeneous preferences. However, learning a mixture model is highly nontrivial, especially when the dataset consists of partial orders. In such cases, the parameter of the model may not be even identifiable. In this paper, we focus on three popular structures of partial orders: ranked top-l1, l2-way, and choice data over a subset of alternatives. We prove that when the dataset consists of combinations of ranked top-l1 and l2-way (or choice data over up to l2 alternatives), mixture of k Plackett-Luce models is not identifiable when l1+l2≤ 2k-1 (l2 is set to 1 when there are no l2-way orders). We also prove that under some combinations, including ranked top-3, ranked top-2 plus 2-way, and choice data over up to 4 alternatives, mixtures of two Plackett-Luce models are identifiable. Guided by our theoretical results, we propose efficient generalized method of moments (GMM) algorithms to learn mixtures of two Plackett-Luce models, which are proven consistent. Our experiments demonstrate the efficacy of our algorithms. Moreover, we show that when full rankings are available, learning from different marginal events (partial orders) provides tradeoffs between statistical efficiency and computational efficiency.