Continuous Correlated Beta Processes

In this paper we consider a (possibly continuous) space of Bernoulli experiments. We assume that the Bernoulli distributions are correlated. All evidence data comes in the form of successful or failed experiments at different points. Current state-ofthe-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, and can be used for both classification 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.