On properties of a new decomposable entropy of DempsterShafer belief functions Abstract   We define entropy of belief functions in the DempsterShafer (DS) theory that satisfies a compound distributions property that is analogous to the property that characterizes Shannonu0027s definitions of entropy and conditional entropy for probability mass functions. None of the existing definitions of entropy for belief functions in the DS theory satisfy this property. We describe some important properties of our definition, and discuss its semantics as a measure of dissonance and not uncertainty. Finally, we compare our definition of entropy with some other definitions that are similar to ours in the sense that these definitions measure dissonance and not uncertainty.

Generalizing Information to the Evolution of Rational Belief Information theory provides a mathematical foundation to measure uncertainty in belief. Belief is represented by a probability distribution that captures our understanding of an outcomexe2x80x99s plausibility. Information measures based on Shannonxe2x80x99s concept of entropy include realization information, Kullbackxe2x80x93Leibler divergence, Lindleyxe2x80x99s information in experiment, cross entropy, and mutual information. We derive a general theory of information from first principles that accounts for evolving belief and recovers all of these measures. Rather than simply gauging uncertainty, information is understood in this theory to measure change in belief. We may then regard entropy as the information we expect to gain upon realization of a discrete latent random variable. This theory of information is compatible with the Bayesian paradigm in which rational belief is updated as evidence becomes available. Furthermore, this theory admits novel measures of information with welldefined properties, which we explored in both analysis and experiment. This view of information illuminates the study of machine learning by allowing us to quantify information captured by a predictive model and distinguish it from residual information contained in training data. We gain related insights regarding feature selection, anomaly detection, and novel Bayesian approaches.