  Claude Shannon coined entropy to quantify the uncertainty of a random
distribution for communication coding theory. We observe that the uncertainty
nature of entropy also limits its direct usage in mathematical modeling.
Therefore we propose a new concept troenpy,as the canonical dual of entropy, to
quantify the certainty of the underlying distribution. We demonstrate two
applications in machine learning. The first is for the classical document
classification, we develop a troenpy based weighting scheme to leverage the
document class label. The second is a self-troenpy weighting scheme for
sequential data and show that it can be easily included in neural network based
language models and achieve dramatic perplexity reduction. We also define
quantum troenpy as the dual of the Von Neumann entropy to quantify the
certainty of quantum systems.
