A simple and efficient Bayesian procedure for selecting dimensionality in multidimensional scaling

Multidimensional scaling (MDS) is a technique which retrieves the locations of objects in a Euclidean space (the object configuration) from data consisting of the dissimilarities between pairs of objects. An important issue in MDS is finding an appropriate dimensionality underlying these dissimilarities. In this paper, we propose a simple and efficient Bayesian approach for selecting dimensionality in MDS. For each column (attribute) vector of an MDS configuration, we assume a prior that is a mixture of the point mass at 0 and a continuous distribution for the rest of the parameter space. Then the marginal posterior distribution of each column vector is also a mixture of the same form, in which the mixing weight of the continuous distribution is a measure of significance for the column vector. We propose an efficient Markov chain Monte Carlo (MCMC) method for estimating the mixture posterior distribution.