Generalized Bayesian Multidimensional Scaling and Model Comparison

Abstract

Multidimensional scaling (MDS) is widely used to reconstruct a low-dimensional representation of high-dimensional data while preserving pairwise distances. However, Bayesian MDS approaches based on Markov chain Monte Carlo (MCMC) face challenges in model generalization and comparison. To address these limitations, we propose a generalized Bayesian multidimensional scaling (GBMDS) framework that accommodates non-Gaussian errors and diverse dissimilarity metrics for improved robustness. We develop an adaptive annealed Sequential Monte Carlo (ASMC) algorithm for Bayesian inference, leveraging an annealing schedule to enhance posterior exploration and computational efficiency. The ASMC algorithm also provides a nearly unbiased marginal likelihood estimator, enabling principled Bayesian model comparison across different error distributions, dissimilarity metrics, and dimensional choices. Using synthetic and real data, we demonstrate the effectiveness of the proposed approach. Our results show that ASMC-based GBMDS achieves superior computational efficiency and robustness compared to MCMC-based methods under the same computational budget. The implementation of our proposed method and applications are available at https://github.com/SFU-Stat-ML/GBMDS.