Continuous Correlated Beta Processes
Robby Goetschalckx, Pascal Poupart and Jesse Hoey
In this paper we consider a (possibly continuous) space of Bernouilli experiments. We assume that the Bernouilli 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 Bernouilli 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 Dirichlet process. This approach can be used for classification (both binary and multi-class) and learning the actual probabilities of the Bernouilli 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 a POMDP model for an automated stroke rehabilitation system.