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- T. Graepel, R. Herbrich, and
K. Obermayer. Bayesian Transduction.
.
In Advances in Neural Information Processing Systems 12, pages
456-462, Cambridge, Massachusetts, 2000. MIT Press.
(FTP Gzipped PostScript, 7 pages, 45 kb)
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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