Hyperbolic Secant representation of the logistic function: Application
  to probabilistic Multiple Instance Learning for CT intracranial hemorrhage
  detection
F. M. Castro-Macías P. Morales-Álvarez Y. Wu R. Molina A. K. Katsaggelos
Multiple Instance Learning (MIL) is a weakly supervised paradigm that has
been successfully applied to many different scientific areas and is
particularly well suited to medical imaging. Probabilistic MIL methods, and
more specifically Gaussian Processes (GPs), have achieved excellent results due
to their high expressiveness and uncertainty quantification capabilities. One
of the most successful GP-based MIL methods, VGPMIL, resorts to a variational
bound to handle the intractability of the logistic function. Here, we formulate
VGPMIL using P\'olya-Gamma random variables. This approach yields the same
variational posterior approximations as the original VGPMIL, which is a
consequence of the two representations that the Hyperbolic Secant distribution
admits. This leads us to propose a general GP-based MIL method that takes
different forms by simply leveraging distributions other than the Hyperbolic
Secant one. Using the Gamma distribution we arrive at a new approach that
obtains competitive or superior predictive performance and efficiency. This is
validated in a comprehensive experimental study including one synthetic MIL
dataset, two well-known MIL benchmarks, and a real-world medical problem. We
expect that this work provides useful ideas beyond MIL that can foster further
research in the field.