Quotient-Based Posterior Analysis for Euclidean Latent Space Models

2026-04-03

AI Summary

This study addresses the non-identifiability of posterior inference in Euclidean latent space models caused by rigid transformations, which renders conventional alignment methods—reliant on arbitrary reference configurations—unstable and difficult to interpret. To overcome this, the authors propose the first theoretically grounded framework for posterior analysis that simultaneously guarantees normalization, existence, and stability. By leveraging quotient space theory and employing a centered Gram map, the approach eliminates non-identifiable degrees of freedom while preserving identifiable geometric structure, enabling intrinsic estimation of posterior means and uncertainties without external alignment. Integrated with Markov chain Monte Carlo post-processing and network latent variable modeling, the method successfully distinguishes stable from reference-sensitive embedding structures in both synthetic data and real-world networks—including the Florentine marriage and statisticians’ collaboration networks—and accurately identifies weakly identified nodes and relationships.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Calibration & Uncertainty Quantification
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Social media analysis through the lenses of networks
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Latent space models are widely used in statistical network analysis and are often fit by Markov chain Monte Carlo. However, posterior summaries of latent coordinates are not canonical because the likelihood depends only on pairwise distances and is invariant under rigid motions of the latent space. Standard post hoc alignment can aid visualization, but the resulting summaries depend on an arbitrary reference configuration. We propose a quotient-based posterior analysis for Euclidean latent space models using the centered Gram map, which represents identifiable latent structure while removing nonidentifiability. This yields intrinsic posterior summaries of mean structure and uncertainty that can be computed directly from posterior samples, together with basic theoretical guarantees including canonicality, existence, and stability. Through simulations and analyses of the Florentine marriage network and a statisticians' coauthorship network, the proposed framework clarifies when alignment-based summaries are stable, when they become reference-sensitive, and which nodes or relationships are weakly identified. These results show how coherent posterior analysis can reveal latent relational structure beyond a single embedding.

Problem

• latent space models
• nonidentifiability
• posterior analysis
• rigid motions
• canonicality

Innovation

• quotient space
• latent space models
• centered Gram map
• posterior analysis
• nonidentifiability