Classically, for decision trees we define a split or various "buckets" to transform continuous data into discrete data. The data I am currently processing has uncertainty associated with it (each data point comes from an aggregate set). As such, I might define a boundary- let's say N, where a data's uncertainty could place it in multiple buckets (say the parameter value N? Normally these boundaries are binary, but I was considering using the probability of these'overlapping instances' towards both buckets weighted by their respective probabilities. This doesn't seem to violate the entropy term (total probability will still sum to 1). However, I can't place half an instance within a branch- which would destroy the meaning behind the term.

Among the many approaches for reasoning about degrees of belief inthe presence of noisy sensing and acting, the logical accountproposed by Bacchus, Halpern, and Levesque is perhaps the most expressive.While their formalism is quite general, it is restricted to fluentswhose values are drawn from discrete countable domains, as opposed tothe continuous domains seen in many robotic applications. In thispaper, we show how this limitation in their approach can be lifted.By dealing seamlessly with both discrete distributions and continuousdensities within a rich theory of action, we provide a very generallogical specification of how belief should change after acting andsensing in complex noisy domains.