Conditional independence and graphical models for rankings

Aritz Pérez

stat.MLNov 3, 2026

Conditional independence and graphical models for rankings

Authors: Aritz Pérez

Organizations: Basque Center for Applied Mathematics (BCAM)

Abstract

Ranking models define probability distributions over rankings and are fundamental in applications such as preference learning and decision-making. Without structural assumptions, these models are intractable due to their factorial complexity. Although independence assumptions are key to reducing complexity and enabling interpretability, standard notions of conditional independence must account for the mutual exclusivity constraints inherent to rankings. We introduce relative rank models, a probabilistic framework based on structured coarsenings of the ranking space that relax these constraints while preserving relative order information. This representation enables a well-defined notion of conditional independence for rankings and allows rankings to be modeled using Bayesian networks. As a result, standard methods for learning and probabilistic inference can be applied to rankings, with interpretable representations based on conditional independence. We further show that two prominent ranking models based on independence assumptions tailored to rankings arise as special cases, providing a unifying framework for independence in ranking models.

Figures & tables

Figure 1: The variable graph

\mathcal{V}=\{(1,2),(2,3)\}

defining

\boldsymbol{X}

for

V=\{1,2,3\}

. The table shows the mapping of

\boldsymbol{X}

from rankings

\sigma\in\mathcal{S}^{V}_{3}

to the corresponding realizations

\boldsymbol{x}\in\boldsymbol{\mathcal{X}}

. Colors in column

\boldsymbol{x}

group the rankings belonging to the same event.

Figure 2: The variable graph

\mathcal{V}=\{(1,2),(2,1),(2,3)\}

contains the cycle

1,2,1

. The realizations

\boldsymbol{x}\in\{(1,1,1),(1,1,2),(2,2,1),(2,2,2)\}

also correspond to empty events and do not appear in the table.

Figure 3: The complete acyclic variable graph

\mathcal{V}=\{(1,2),(2,3),(1,3)\}

, where

\bar{\pi}=[1,2,3]

is its unique ancestral ordering. The RRV defined by

\mathcal{V}

induces a thin partition of the ranking space.

Appendix figures & tables2 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 4 — Figure 4: Five variable graphs defined over items $V=\{1,2,3\}$ .

		$\mathcal{V}_{1}$	$\mathcal{V}_{2}$	$\mathcal{V}_{3}$	$\mathcal{V}_{4}$	$\mathcal{V}_{5}$
$\sigma$	$\bar{\sigma}$	$\boldsymbol{x}=(x_{1},x_{2},x_{3})$
1,2,3	1,2,3	1,1,2	1,2,1	1,2,2	1,2,2	1,1,1
1,3,2	1,3,2	1,1,1	1,3,1	1,2,2	1,2,1	1,2,1
2,1,3	2,1,3	2,1,2	1,1,1	1,1,2	1,1,2	2,1,1
2,3,1	3,1,2	1,1,1	1,3,1	1,2,1	2,2,1	2,2,1
3,1,2	2,3,1	2,1,2	1,1,1	1,1,1	2,1,2	3,1,1

Appendix

Table 1: Relative rank variable realizations

\boldsymbol{x}=(x_{1},x_{2},x_{3})

for the rankings

\sigma\in\mathcal{S}^{V}_{n}

and orderings

\bar{\sigma}\in\mathcal{S}^{n}_{V}

, defined according to the five variable graphs shown in Figure 4 . The graph

\mathcal{V}_{4}

is cyclic and the realizations

\boldsymbol{x}\in\{(1,1,1),(2,2,2)\}

do not appear in the table because they correspond to empty events.

Probabilistic Graphical Models

Probabilistic graphical models represent joint distributions with graphs to support structured inference, learning, and reasoning under uncertainty.

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Population-Level Generative Modeling for Ranking Data

Zhaoyang Shi

Center for Applied Mathematics, Fudan University Shanghai, China

Synthetic Data Generation

Open paper AskPDF arXiv

Ranking data arise in scientific and machine learning applications, including recommendation systems, information retrieval, voting, marketing, and AI preference ranking from human feedback. Existing statistical work has primarily focused on inference tasks such as preference estimation, rank aggregation, and ranking prediction. However, generating realistic synthetic rankings from an observed population is important for privacy-preserving data sharing, benchmark construction, simulation, and uncertainty quantification. This task is challenging because rankings are high-dimensional combinatorial objects with non-Euclidean dependence structures, while ranking populations often exhibit substantial preference heterogeneity. We propose a framework for population-level generative modeling through a latent preference simplex embedding. It estimates a low-dimensional latent preference simplex through a likelihood-based ranking model, leverages flow matching to learn the population distribution of latent preferences, and generates new rankings through the fitted probabilistic ranking model. We show that ranking generation admits an oracle reduction to latent distribution learning and derive finite-sample generative guarantees that clarify how the number of items, ranking length, and latent dimension affect accuracy. Experiments on synthetic and real datasets demonstrate improved population-level fidelity and provide a statistically interpretable representation of preference heterogeneity.

Abstract

Probabilistic Graphical Models

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