Generalized entropy optimized by an arbitrary distribution

We construct the generalized entropy optimized by a given arbitrary statistical distribution with a finite linear expectation value of a random quantity of interest. This offers, via the maximum entropy principle, a unified basis for a great variety of distributions observed in nature, which can hardly be described by the conventional methods. As a simple example, we explicitly derive the entropy associated with the stretched exponential distribution. To include the distributions with the divergent moments (e.g., the Levy stable distributions), it is necessary to modify the definition of the expectation value.Comment: 10 pages, no figure

From Laplace to supernova SN 1987a: Bayesian Inference In Astrophysics

The Bayesian approach to probability theory is presented as an alternative to the currently used long-run relative frequency approach, which does not offer clear, compelling criteria for the design of statistical methods. Bayesian probability theory offers unique and demonstrably optimal solutions to well-posed statistical problems, and is historically the original approach to statistics. The reasons for earlier rejection of Bayesian methods are discussed, and it is noted that the work of Cox, Jaynes, and others answers earlier objections, giving Bayesian inference a firm logical and mathematical foundation as the correct mathematical language for quantifying uncertainty. The Bayesian approaches to parameter estimation and model comparison are outlined and illustrated by application to a simple problem based on the gaussian distribution. As further illustrations of the Bayesian paradigm, Bayesian solutions to two interesting astrophysical problems are outlined: the measurement of weak signals in a strong background, and the analysis of the neutrinos detected from supernova SN 1987A. A brief bibliography of astrophysically interestin