Bayesian Metric Multidimensional Scaling

Abstract

In this article, we show how to apply Bayesian methods to noisy ratio scale distances for both the classical similarities problem as well as the unfolding problem. Bayesian methods produce essentially the same point estimates as the classical methods, but are superior in that they provide more accurate measures of uncertainty in the data. Identification is nontrivial for this class of problems because a configuration of points that reproduces the distances is identified only up to a choice of origin, angles of rotation, and sign flips on the dimensions. We prove that fixing the origin and rotation is sufficient to identify a configuration in the sense that the corresponding maxima/minima are inflection points with full-rank Hessians. However, an unavoidable result is multiple posterior distributions that are mirror images of one another. This poses a problem for Markov chain Monte Carlo (MCMC) methods. The approach we take is to find the optimal solution using standard optimizers. The configuration of points from the optimizers is then used to isolate a single Bayesian posterior that can then be easily analyzed with standard MCMC methods.