Bayesian Distance-to-Set Models: from Latent Variable to Latent Projection

Abstract

Statistical models often assume that data are generated near a structured, smooth, or low-dimensional set. A common approach is to use Bayesian latent variable models, in which each observation is associated with a latent coordinate on the set, and the observed data are modeled as noisy deviations from these coordinates. The deviation is typically characterized by a location-scale distribution, such as Gaussian. Despite their intuitive appeal and popularity, latent variable models often present practical challenges in posterior computation. In particular, Markov chain Monte Carlo samplers may suffer from slow mixing, especially when the sample size is large and there is no closed form for integrating out the latent coordinates. In this article, we propose an alternative approach that replaces the deviation-from-coordinate with a distance-to-set. Specifically, the distance-to-set is defined as the distance between a data point and its projection onto the set, where the projection can be rapidly computed by optimization and replaces the latent coordinate in the likelihood. This change substantially reduces the dimensionality of the parameter X latent variable space, leading to efficient posterior computation. We establish several important statistical properties for the distance-to-set models, such as the independence between the normal-cone noise and fixed-effect parameters, posterior consistency, and an Occam's razor effect that automatically penalizes overfitting. We demonstrate the effectiveness of our approach through simulation studies, applications to multi-environment study and Bayesian transfer learning.