Contextual Probability Estimation from Data Samples – A Generalisation

Abstract

Contextual probability (G) provides an alternative, efficient way of estimating (primary) probability (P) in a principled way. G is defined in terms of P in a combinatorial way, and they have a simple linear relationship. Consequently, if one is known, the other can be calculated. It turns out G can be estimated based on a set of data samples through a simple process called neighbourhood counting. Many results about contextual probability are obtained based on the assumption that the event space is the power set of the sample space. However, the real world is usually not the case. For example, in a multidimensional sample space, the event space is typically the set of hyper tuples which is much smaller than the power set. In this paper, we generalise contextual probability to multidimensional sample space where the attributes may be categorical or numerical. We present results about the normalisation constant, the relationship between G and P and the neighbourhood counting process.