I remember a paper from 1999 (13 years ago!) called Probabilistic Outputs for Support Vector Machines and Comparisons to Regularized Likelihood Methods (1999) by John Platt that outlined a method for getting probabilistic outputs out of an SVM. From the abstract:
Instead, we train an SVM, then train the parameters of an additional sigmoid function to map the SVM outputs into probabilities.
Whilst this provides a solution of sorts, it is slightly unsatisfactory as it means performing two separate (and seemingly somewhat unrelated) optimisation problems.
Is there a more modern approach to this problem (without resorting to Gaussian Process classification for example)?