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Abstract:
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.
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