Entropy and Extropy for Partial Probability Assessments on Arbitrary Families of Events

In subjective probability theory, a coherent probability assertion represents an honest expression of its promoter’s uncertain knowledge about the value of an unknown quantity. The theory of proper scoring rules was central to de Finetti’s ideas about assessing the relative values of different subjective probability assessments. In this paper, we consider an asymmetric proper scoring rule for the probability of an event, which belongs to the 2-parameter Beta family of scoring rules. Then we consider the associated loss function of a probability assessment on an arbitrary family of n events. We observe that, in the particular case of a probability mass distribution of a random quantity, the expected loss function associated with the asymmetric score coincides with the Shannon entropy. Likewise, we show similar properties between the notion of extropy and the expected loss function associated with the complement of the asymmetric score. Then, we suitably extend the notion of entropy and extropy from partitions of events to arbitrary families of events. We also introduce Bregman divergences associated with these measures of information. Finally, we introduce a symmetric proper scoring rule for an event, showing that the associated expected loss function for an arbitrary family of events coincides with the sum of entropy and extropy.