Search results for "Scoring rule"
showing 9 items of 9 documents
Extropy: Complementary Dual of Entropy
2015
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…
Approximations in Statistics from a Decision-Theoretical Viewpoint
1987
The approximation of the probability density p(.) of a random vector x∊X by another (possibly more convenient) probability density q(.) which belongs to a certain class Q is analyzed as a decision problem where the action space is the class Qof available approximations, the relevant uncertain event is the actual value of the vector x and the utility function is a proper scoring rule. The logarithmic divergence is shown to play a rather special role within this approach. The argument lies entirely within a Bayesian framework.
Assessing fat-tailed sequential forecast distributions for the Dow-Jones index with logarithmic scoring rules
2007
We use the logarithmic scoring rule for distributions to assess a variety of fat-tailed sequential forecasting distributions for the Dow-Jones industrial stock index from 1980 to the present. The methodology applies Bruno de Finetti''s contributions to understanding how to compare the quality of different coherent forecasting distributions for the same sequence of observations, using proper scoring rules. Four different forms of forecasting distributions are compared: a mixture Normal, a mixture of convex combinations of three Normal distributions, a mixture exponential power distribution, and a mixture of a convex combination of three exponential power distributions. The mixture linear com…
The Duality of Entropy/Extropy, and Completion of the Kullback Information Complex
2018
The refinement axiom for entropy has been provocative in providing foundations of information theory, recognised as thoughtworthy in the writings of both Shannon and Jaynes. A resolution to their concerns has been provided recently by the discovery that the entropy measure of a probability distribution has a dual measure, a complementary companion designated as &ldquo
Sequentially Forecasting Economic Indices Using Mixture Linear Combinations of EP Distributions
2021
This article displays an application of the statistical method moti- vated by Bruno de Finetti's operational subjective theory of probability. We use exchangeable forecasting distributions based on mixtures of linear com- binations of exponential power (EP) distributions to forecast the sequence of daily rates of return from the Dow-Jones index of stock prices over a 20 year period. The operational subjective statistical method for comparing distributions is quite different from that commonly used in data analysis, because it rejects the basic tenets underlying the practice of hypothesis test- ing. In its place, proper scoring rules for forecast distributions are used to assess the values o…
Completing the logarithmic scoring rule for assessing probability distributions
2012
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 f…
SCORING ALTERNATIVE FORECAST DISTRIBUTIONS: COMPLETING THE KULLBACK DISTANCE COMPLEX
2018
We develop two surprising new results regarding the use of proper scoring rules for evaluating the predictive quality of two alternative sequential forecast distributions. Both of the proponents prefer to be awarded a score derived from the other's distribution rather than a score awarded on the basis of their own. A Pareto optimal exchange of their scoring outcomes provides the basis for a comparison of forecast quality that is preferred by both forecasters, and also evades a feature of arbitrariness inherent in using the forecasters' own achieved scores. The well-known Kullback divergence, used as a measure of information, is evaluated via the entropies in the two forecast distributions a…
Coherent Conditional Previsions and Proper Scoring Rules
2012
In this paper we study the relationship between the notion of coherence for conditional prevision assessments on a family of finite conditional random quantities and the notion of admissibility with respect to bounded strictly proper scoring rules. Our work extends recent results given by the last two authors of this paper on the equivalence between coherence and admissibility for conditional probability assessments. In order to prove that admissibility implies coherence a key role is played by the notion of Bregman divergence.
Coherent conditional probabilities and proper scoring rules
2011
In this paper we study the relationship between the notion of coherence for conditional probability assessments on a family of conditional events and the notion of admissibility with respect to scoring rules. By extending a recent result given in literature for unconditional events, we prove, for any given strictly proper scoring rule s, the equivalence between the coherence of a conditional probability assessment and its admissibility with respect to s. In this paper we focus our analysis on the case of continuous bounded scoring rules. In this context a key role is also played by Bregman divergence and by a related theoretical aspect. Finally, we briefly illustrate a possible way of defin…