Bayesian Transduction

Transduction is an inference principle that takes a training sample and aims at estimating the values of a function at given points contained in the so-called working sample. Hence, transduction is a less ambitious task than induction which aims at inferring a functional dependency on the whole of input space. As a consequence, however, transduction provides a confidence measure on single predictions rather than classifiers, a feature particularly important for risk–sensitive applications. We consider the case of binary classification by linear discriminant functions (perceptrons) in kernel space. From the transductive point of view, the infinite number of perceptrons is boiled down to a finite number of equivalence classes on the working sample each of which corresponds to a polyhedron in parameter space. In the Bayesian spirit the posteriori probability of a labelling of the working sample is determined as the ratio between the volume of the corresponding polyhedron and the volume of version space. Then the maximum posteriori scheme recommends to choose the labelling of maximum volume. We suggest to sample version space by an ergodic billiard in kernel space. Experimental results on real world data indicate that Bayesian Transduction compares favourably to the well-known Support Vector Machine, in particular if the posteriori probability of labellings is used as a confidence measure to exclude test points of low confidence.