Wallenius Bayes

This paper introduces a new event model appropriate for classifying (binary) data generated by a “destructive choice” process, such as certain human behavior. In such a process, making a choice removes that choice from future consideration yet does not influence the relative probability of other choices in the choice set. The proposed Wallenius event model is based on a somewhat forgotten non-central hypergeometric distribution introduced by Wallenius (Biased sampling: the non-central hypergeometric probability distribution. Ph.D. thesis, Stanford University, 1963). We discuss its relationship with models of how human choice behavior is generated, highlighting a key (simple) mathematical property. We use this background to describe specifically why traditional multivariate Bernoulli naive Bayes and multinomial naive Bayes each are suboptimal for such data. We then present an implementation of naive Bayes based on the Wallenius event model, and show experimentally that for data where we would expect the features to be generated via destructive choice behavior Wallenius Bayes indeed outperforms the traditional versions of naive Bayes for prediction based on these features. Furthermore, we also show that it is competitive with non-naive methods (in particular, support-vector machines). In contrast, we also show that Wallenius Bayes underperforms when the data generating process is not based on destructive choice.