A probabilistic framework for the design of instance-based supervised ranking algorithms in an ordinal setting

In this article, we present a probabilistic framework which serves as the base from which instance-based algorithms for solving the supervised ranking problem may be derived. This framework constitutes a simple and novel approach to the supervised ranking problem, and we give a number of typical examples of how this derivation can be achieved. In this general framework, we pursue a cumulative and stochastic approach, relying heavily upon the concept of stochastic dominance. We show how the median can be used to extract, in a consistent way, a single (classification) label from a returned cumulative probability distribution function. We emphasize that all operations used are mathematically sound, i.e. they only make use of ordinal properties. Mostly, when confronted with the problem of learning a ranking, the training data is not monotone in itself, and some cleansing operation is performed on it to remove these 'inconsistent' examples. Our framework, however, deals with these occurrences of 'reversed preference' in a non-invasive way. On the contrary, it even allows to incorporate information gained from the occurrence of these reversed preferences. This is exactly what happens in the second realization of the main theorem.