Abstract
While Shannon’s differential entropy adequately quantifies a dimensioned random variable’s information deficit under a given measurement system, the same cannot be said of differential weighted entropy in its existing formulation. We develop weighted and residual weighted entropies of a dimensioned quantity from their discrete summation origins, exploring the relationship between their absolute and differential forms, and thus derive a “differentialized” absolute entropy based on a chosen “working granularity” consistent with Buckingham’s -theorem. We apply this formulation to three common continuous distributions: exponential, Gaussian, and gamma and consider policies for optimizing the working granularity.
Highlights
Informational entropy, introduced by Shannon [1] as an analogue of the thermodynamic concept developed by Boltzmann and Gibbs [2], represents the expected information deficit prior to an outcome selected from a set or range of possibilities with known probabilities
Many modern applications using this concept have been developed, such as the so-called maximum entropy method for choosing the “best yet simplest” probabilistic model from amongst a set of parameterized models, which is statistically consistent with observed data
Some authors call fX (x)/FX (t) the “hazard function” though this is only valid for the case of x = t; it is better interpreted as the probability density function (PDF) of X subject to the condition X ≥ t
Summary
Informational entropy, introduced by Shannon [1] as an analogue of the thermodynamic concept developed by Boltzmann and Gibbs [2], represents the expected information deficit prior to an outcome (or message) selected from a set or range of possibilities with known probabilities. This means that we find the parameter values which maximize the entropy of the model, subject to constraints that ensure the model is consistent with the observed data, and MacKay [5] has given a Bayesian probabilistic explanation for the basis of Occam’s razor This maximum entropy approach has found widespread applications in image processing to reconstruct images from noisy data [6,7] for example, in Astronomy, where signal to noise levels are often extremely low - and in speech and language processing, including automatic speech recognition and automated translation [3,8]. We further propose and evaluate a potential remedy for this problem; namely a finite working granularity
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
