Generalized Bayesian multidimensional scaling and model comparison
摘要
Multidimensional scaling (MDS) is a fundamental dimension reduction technique widely applied across diverse fields to represent objects as points in a lower-dimensional space based on pairwise dissimilarities. While classical and scalable non-Bayesian MDS methods are valued for their computational efficiency, they primarily focus on optimization, yielding only point estimates and thus neglecting model uncertainty. This limitation motivates the development of Bayesian MDS (BMDS) frameworks, which offer a probabilistic perspective, model uncertainty, and provide posterior distributions for principled uncertainty quantification. However, existing BMDS methods predominantly assume Euclidean distance and Gaussian noise in the observed dissimilarities, which limits robustness and applicability across diverse domains like text mining. Furthermore, Bayesian inference for these models faces computational challenges when scaling to large datasets. To overcome these issues, we propose a Generalized Bayesian Multidimensional Scaling (GBMDS) framework that incorporates flexible dissimilarity metrics and robust non-Gaussian error structures, prioritizing uncertainty quantification, robustness, and model flexibility. To facilitate efficient inference and robust model selection within this framework, we design an adaptive annealed Sequential Monte Carlo (ASMC) algorithm. This ASMC approach mitigates computational burdens in large-scale applications, leverages existing MCMC proposals for ease of implementation, and provides nearly unbiased estimators of marginal likelihoods, enabling rigorous Bayesian model comparison via Bayes factors. The GBMDS framework thus enhances estimation accuracy and robustness by accommodating complex dissimilarities and heavy-tailed distributions.