This article provides a completion to theories of information based on entropy, resolving a longstanding question in its axiomatization as proposed by Shannon and pursued by Jaynes. We show that Shannon's entropy function has a complementary dual function which we call "extropy." The entropy and the extropy of a binary distribution are identical. However, the measure bifurcates into a pair of distinct measures for any quantity that is not merely an event indicator. As with entropy, the maximum extropy distribution is also the uniform distribution, and both measures are invariant with respect to permutations of their mass functions. However, they behave quite differently in their assessments of the refinement of a distribution, the axiom which concerned Shannon and Jaynes. Their duality is specified via the relationship among the entropies and extropies of course and fine partitions. We also analyze the extropy function for densities, showing that relative extropy constitutes a dual to the Kullback-Leibler divergence, widely recognized as the continuous entropy measure. These results are unified within the general structure of Bregman divergences. In this context they identify half the $L_2$ metric as the extropic dual to the entropic directed distance. We describe a statistical application to the scoring of sequential forecast distributions which provoked the discovery.Comment: Published at http://dx.doi.org/10.1214/14-STS430 in the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org

During the past decade, Hector's dolphins, Cephalorhynchus hectori, have suffered an alarming level of mortality due to entanglement in commercial and amateur gill nets. In this paper we study two Leslie matrix population models that incorporate known features of dolphin fertility and mortality, focussing on the information they provide regarding age distributions and maximum population growth rates. The simplest model specifies constant survival rates over many age-classes. The second model uses more realistic curves of age-specific survival rates. The results indicate that Hector's dolphin, like most other small cetaceans, has a low potential for population growth. Growth rates of 1.8–4.9% per year are likely to be the maximum possible for Hector's dolphin populations, and C. hectori (and C. commersonii) populations are likely to be declining under recent levels of net entanglement. Survival rate estimates from free-living populations, subject to natural and net-entanglement mortality, showed decreasing populations. Even with the most optimistic reproductive parameters, survival rates would need to be some 5–10% higher than those observed in populations subject to gill-net entanglement before population growth could occur. The likely consequences of a reduction in entanglement mortality through conservation management are explored using the survivorship curve model. These simulations show that the age structure of the population can have an important effect on changes in the size and growth rate of the population during the recovery phase following a reduction in entanglement mortality.

Re‐sightings of photographically identified individuals were used to estimate survival rates for a free‐living population of Hector's dolphins Cephalorhynchus hectori, a species endemic to New Zealand waters. Most individuals were identified from injuries to the dorsal fin. Consequently, the photographic catalog contained very few young individuals. Our analysis included no newborn calves or yearlings, and provided estimates of survival rates only after the first year of life. We used two complementary methods for calculating survival rates: a modified Jolly‐Seber model, and a simpler method which corrects in a more explicit way for individual dolphins being alive but not sighted. Selection of the most reliable subset of the data had a greater effect on computed survival rates than did the difference between the two methods. We conclude that careful inspection of resighting data before analysis, and, if necessary, selection of a subset, is very important in studies of this kind. Survival rate estimates came from a population which was subject to relatively heavy mortality from gillnet entanglement. Standard errors of the survival rate estimates have been used to assess the conditional probability of population decline given three fertility scenarios. The high probability that the Banks Peninsula Hector's dolphin population was decreasing during the study period (0.78 to 0.99) suggests that gillnet entanglement constituted a serious risk to this population.

We display the first two moment functions of the Logitnormal family of distributions, conveniently described in terms of the Normal mean, m, and the Normal signal-to-noiseratio, m/s, parameters that generate the family. Long neglected on account of the numerical integrations required to compute them, awareness of these moment functions should aid the sensible interpretation of logistic regression statistics and the specification of 'diffuse' prior distributions in hierarchical models, which can be deceiving. We also use numerical integration to compare the correlation between bivariate Logitnormal variables with the correlation between the bivariate Normal variables from which they are transformed

We show how Bruno de Finetti''s fundamental theorem of prevision has computable applications in statistical problems that involve only partial information. Specifically, we assess accuracy rates for median decision procedures used in the radiological diagnosis of asbestosis. Conditional exchangeability of individual radiologists'' diagnoses is recognized as more appropriate than independence which is commonly presumed. The FTP yields coherent bounds on probabilities of interest when available information is insufficient to determine a complete distribution. Further assertions that are natural to the problem motivate a partial ordering of conditional probabilities, extending the computation from a linear to a quadratic programming problem