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

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Swimming pool activities revealed to be efficacious to train psychomotor skills and increase adaptive behaviors in children with Autism Spectrum Disorders (ASD). Therefore, the purpose of this study was to investigate the efficacy of a specific multi-systemic aquatic therapy (CI-MAT) on gross motor and social skills in three adolescents with Autism Spectrum Disorders (ASD). Methods: three adolescents with ASD of which two boys (M1 with a chronological age of 10.3 years and a mental age of 4.7 years; M2 with a chronological age of 14.6 and a mental age inferior to 4 years) and one girl (chronological age of 14.0 and a mental age inferior to 4 years). The study was divided into three phases: baseline, 12-week CI-MAT program and Post-Test. Participants were administered a battery of tests incorporating anthropometric measurements, gross motor development test and a social skills questionnaire before and after a 12-week MAT-CI program. Results: Subjects improved locomotors and object control skills following the CI-MAT program in a different way. Concerning social behaviors, the higher proportion of gains was observed in the sensitivity of other’s presence and eye contact, for the contact domain, and in the comply turn for the relationship domain. Conclusions: The results of this study showed that the CI-MAT program was effective for the development of gross-motor skills and social behaviors in subjects with ASD. Moreover there is an urge to carry out a whole psychological assessment targeting both motor and adaptive development suitable to provide educational and vocational plans of exercises for people with ASD.

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We propose and motivate an expanded version of the logarithmic score for forecasting distributions, termed the Total Log score. It incorporates the usual logarithmic score, which is recognised as incomplete and has been mistakenly associated with the likelihood principle. The expectation of the Total Log score equals the Negentropy plus the Negextropy of the distribution. We examine both discrete and continuous forms of the scoring rule, and we discuss issues of scaling for scoring assessments. The analysis suggests the dual tracking of the quadratic score along with the usual log score when assessing the qualities of probability distributions. An application to the sequential scoring of forecast distributions for the daily rate of stock returns displays the usefulness of the proposal