Inductive Inference and Partition Exchangeability in Classification

Inductive inference has been a subject of intensive research efforts over several decades. In particular, for classification problems substantial advances have been made and the field has matured into a wide range of powerful approaches to inductive inference. However, a considerable challenge arises when deriving principles for an inductive supervised classifier in the presence of unpredictable or unanticipated events corresponding to unknown alphabets of observable features. Bayesian inductive theories based on de Finetti type exchangeability which have become popular in supervised classification do not apply to such problems. Here we derive an inductive supervised classifier based on partition exchangeability due to John Kingman. It is proven that, in contrast to classifiers based on de Finetti type exchangeability which can optimally handle test items independently of each other in the presence of infinite amounts of training data, a classifier based on partition exchangeability still continues to benefit from a joint prediction of labels for the whole population of test items. Some remarks about the relation of this work to generic convergence results in predictive inference are also given.