Continuous Correlated Beta Processes

Abstract

In this paper we consider a (possibly continuous) space of Bernoulli experiments. We assume that the Bernoulli distributions of the points are correlated. All evidence data comes in the form of successful or failed experiments at different points. Current state-of-the-art methods for expressing a distribution over a continuum of Bernoulli distributions use logistic Gaussian processes or Gaussian copula processes. However, both of these require computationally expensive matrix operations (cubic in the general case). We introduce a more intuitive approach, directly correlating beta distributions by sharing evidence between them according to a kernel function, an approach which has linear time complexity. The approach can easily be extended to multiple outcomes, giving a continuous correlated Dirichlet process.This approach can be used for classification (both binary and multi-class) and learning the actual probabilities of the Bernoulli distributions. We show results for a number of data sets, as well as a case-study where a mixture of continuous beta processes is used as part of an automated stroke rehabilitation system.