Bayesian Inference for Multidimensional Scaling Representations with Psychologically Interpretable Metrics

Abstract

Multidimensional scaling (MDS) models represent stimuli as points in a space consisting of a number of psychological dimensions, such that the distance between pairs of points corresponds to the dissimilarity between the stimuli. Two fundamental challenges in inferring MDS representations from data involve inferring the appropriate number of dimensions and the metric structure of the space used to measure distance. We approach both challenges as Bayesian model-selection problems. Treating MDS as a generative model, we define priors needed for model identifiability under metrics corresponding to psychologically separable and psychologically integral stimulus domains. We then apply a differential evolution Markov-chain Monte Carlo (DE-MCMC) method for parameter inference, and a Warp-III method for model selection. We apply these methods to five previous data sets, which collectively test the ability of the methods to infer an appropriate dimensionality and to infer whether stimuli are psychologically separable or integral. We demonstrate that our methods produce sensible results, but note a number of remaining technical challenges that need to be solved before the method can easily and generally be applied. We also note the theoretical promise of the generative modeling perspective, discussing new and extended models of MDS representation that could be developed.