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## Weighted Entropy and its Use in Computer Science and Beyond

P. 293-308.

The concept of weighted entropy takes into account values of

different outcomes, i.e., makes entropy context-dependent, through the

weight function. We analyse analogs of the Fisher information inequality

and entropy-power inequality for the weighted entropy and discuss connections

with weighted Lieb’s splitting inequality. The concepts of rates

of the weighted entropy and information are also discussed.

Publication based on the results of:

### In book

Vol. 10684: Analytical and Computational Methods in Probability Theory. , Berlin : Springer, 2017

Porto : MDPI Open Access Publishing, 2021

Added: May 5, 2021

Pianykh O., Pospelova K., Kamboj N., Journal of Digital Imaging 2018 Vol. 31 No. 6 P. 768-775

Humans can determine image quality instantly and intuitively, but the mechanism of human perception of image quality is unknown. The purpose of this work was to identify the most important quantitative metrics responsible for the human perception of digital image quality. Digital images from two different datasets—CT tomography (MedSet) and scenic photographs of trees (TreeSet)—were ...

Added: February 5, 2019

Kolesnikov A., Теория вероятностей и ее применения 2012 Т. 57 № 2 С. 296-321

We study Sobolev a priori estimates for the optimal transportation $T = \nabla \Phi$ between probability measures $\mu=e^{-V} \ dx$ and $\nu=e^{-W} \ dx$ on $\R^d$.
Assuming uniform convexity of the potential $W$ we show that $\int \| D^2 \Phi\|^2_{HS} \ d\mu$, where $\|\cdot\|_{HS}$ is the Hilbert-Schmidt norm,
is controlled by the Fisher information of $\mu$. In ...

Added: February 19, 2013

Kelbert M., Mozgunov P., Mathematical Communications 2015 Vol. 20 No. 2 P. 219-228

We study the asymptotic behaviour of differential entropy in a Bayesian problem of estimationg a probability of success in a series
of conditionally independent trials. ...

Added: October 19, 2015

Kelbert M., Mozgunov P., Mathematical Communications 2017 Vol. 22 No. 1 P. 25-40

The paper considers a family of probability distributions depending on a parameter. The goal is to derive the generalized versions of Cram´er-Rao and Bhattacharyya
inequalities for the weighted covariance matrix and of the Kullback inequality for the
weighted Kullback distance, which are important objects themselves [9, 23, 28]. The asymptotic forms of these inequalities for a particular ...

Added: March 10, 2017

Kosov E., Journal of Mathematical Analysis and Applications 2021 Vol. 504 No. 1 Article 125358

The paper studies the sampling discretization problem for integral norms on subspaces of L^p(\mu). Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector ...

Added: October 1, 2021

Erokhina Y., International Journal for the Semiotics of Law 2022 Vol. 35 No. 3 P. 1187-1213

Fake news is created as ordinary news stylistically but it consists of deliberate disinformation or hoaxes (aimed at misinforming or deceiving people). The text is generally constructed to cause negative emotions and feelings in readers: fear, panic, distrust, and paranoia. It is done to manipulate the opinion and consciousness of a large number of people ...

Added: February 23, 2022

Mozgunov P., Kelbert M., , in : Proceedings of Information Technology and Systems 2015. : Sochi : ., 2015. P. 614-621.

Consider a Bayesian problem of estimating of probability of success in a series of trials with binary outcomes. We study the asymp- totic behaviour of weighted differential entropy for posterior probability density function (PDF) conditional on x successes after n trials, when n → ∞. Suppose that one is interested to know whether the coin is fair or not ...

Added: December 5, 2015

Popkov Y., 2012 No. 14 P. 924-944

published ...

Added: November 14, 2013

Polyvyanyy A., Kalenkova A. A., , in : 2019 International Conference on Process Mining (ICPM). : IEEE, 2019. P. 81-88.

Conformance checking is a subarea of process mining that studies relations between designed processes, also called process models, and records of observed processes, also called event logs. In the last decade, research in conformance checking has proposed a plethora of techniques for characterizing the discrepancies between process models and event logs. Often, these techniques are ...

Added: October 13, 2019

Mozgunov P., Kelbert M., Eurasian Mathematical Journal 2015 Vol. 6 No. 2 P. 6-17

Consider a Bayesian problem of success probability estimation in a series of conditionally independent trials with binary outcomes. We study the asymptotic behaviour of the weighted differential entropy for posterior probability density function conditional on x successes after n conditionally independent trials when n tends to infinity. Suppose that one is interested to know whether ...

Added: December 5, 2015

Di Gesù G., Mariani M., SIAM Journal on Mathematical Analysis 2017 Vol. 49 No. 4 P. 3048-3072

We establish an expansion by Gamma-convergence of the Fisher information relative to the reference measure e (beta V)dx, where V is a generic multiwell potential and beta -> infinity. The expansion reveals a hierarchy of scales reflecting the metastable behavior of the underlying overdamped Langevin dynamics: distinct scales emerge and become relevant depending on whether ...

Added: October 8, 2018

Koltsov S., Письма в Журнал технической физики 2017 Т. 43 № 12 С. 90-95

A thermodynamic approach has been applied to solving the problem of selecting the number of
clusters/topics in topic modeling. The main principles of this approach are formulated and the behavior of
topic models during temperature variations is studied. Using thermodynamic formalism, the existence of the
entropy phase transition in topic models is shown and criteria for the choice ...

Added: April 18, 2017

Apenko S.M., Physica A: Statistical Mechanics and its Applications 2014 Vol. 414 P. 108-114

We discuss a possibility of deriving an H-theorem for nonlinear discrete time evolution
equation that describes random wealth exchanges. In such kinetic models economical
agents exchange wealth in pairwise collisions just as particles in a gas exchange their energy.
It appears useful to reformulate the problem and represent the dynamics as a combination
of two processes. The first is ...

Added: October 23, 2014

Kolokoltsov V., Transactions of the Moscow Mathematical Society 2021 Vol. 82 No. 1 P. 93-104

Combining intuitive probabilistic assumptions with the basic laws of classical thermodynamics, using
the latter to express probabilistic parameters in terms of the thermodynamic quantities, we get a simple
unified derivation of the fundamental ensembles of statistical physics avoiding any limiting procedures,
quantum hypothesis and even statistical entropy maximization. This point of view leads also to some
related classes of ...

Added: October 29, 2021

Kovalev A., Energy 2016 Vol. 100 P. 129-136

The direct relationship between thermodynamic entropy and economic scarcity is only valid for a thermodynamically isolated economy. References to the second law of thermodynamics in economics within the context of scarcity ignore the fact that the earth is not an isolated system. The earth interacts with external sources and sinks of entropy and the resulting ...

Added: February 19, 2016

Kolesnikov A., Kosov E., Theory of Stochastic Processes 2017 Vol. 22 No. 38 P. 47-61

Let γ be the standard Gaussian measure on Rn and let Pγ be the space of probability measures that are absolutely continuous with respect to γ. We study lower bounds for the functional Fγ(µ) = Ent(µ) − 1 2W2 2 (µ, ν), where µ ∈ Pγ, ν ∈ Pγ, Ent(µ) = R log µ γ ...

Added: August 21, 2018

Moulin H., Journal of Economic Theory 2016 Vol. 162 P. 1-20

An assignment of students to schools displays zero-segregation if all schools have the same distribution of the different types of students (ethnic, geographical, gender). We axiomatize the choice of an optimally desegregated assignment under arbitrary capacity constraints. The celebrated Consistency axiom, together with standard rational choice requirements, identify the choice rule minimizing a canonical index ...

Added: April 12, 2016