article

Maximum Entropy Principle with General Deviation Measures

Authors:

Bogdan Grechuk,

Anton Molyboha,

Michael ZabarankinAuthors Info & Claims

Mathematics of Operations Research, Volume 34, Issue 2

Pages 445 - 467

https://doi.org/10.1287/moor.1090.0377

Published: 01 May 2009 Publication History

Abstract

An approach to the Shannon and Rényi entropy maximization problems with constraints on the mean and law-invariant deviation measure for a random variable has been developed. The approach is based on the representation of law-invariant deviation measures through corresponding convex compact sets of nonnegative concave functions. A solution to the problem has been shown to have an alpha-concave distribution (log-concave for Shannon entropy), for which in the case of comonotone deviation measures, an explicit formula has been obtained. As an illustration, the problem has been solved for several deviation measures, including mean absolute deviation (MAD), conditional value-at-risk (CVaR) deviation, and mixed CVaR-deviation. Also, it has been shown that the maximum entropy principle establishes a one-to-one correspondence between the class of alpha-concave distributions and the class of comonotone deviation measures. This fact has been used to solve the inverse problem of finding a corresponding comonotone deviation measure for a given alpha-concave distribution.

References

[1]

Artzner, P., Delbaen, F., Eber, J.-M. and Heath, D., "Coherent measures of risk," Math. Finance, v9, pp. 203-227, 1999.

[2]

Bialas, J. and Nakamura, Y., "The theorem of Weierstrass," J. Formalized Math., v5, pp. 353-359, 1996.

[3]

Benoist, J. and Daniilidis, A., "Coincidence theorems for convex functions," J. Convex Anal., v9, pp. 259-268, 2002.

[4]

Brody, D. C., Buckley Ian, R. C. and Constantinou, I. C., "Option price calibration from Rényi entropy," Phys. Lett. A, v366, pp. 298-307, 2007.

[5]

Buchen, P. and Kelly, M., "The maximum entropy distribution of an asset inferred from option prices," J. Financial Quant. Anal., v31, pp. 143-159, 1996.

[6]

Costa, J., Hero, A., Vignat, C., Rangarajan, A., Figueiredo, M. and Zerubia, J., "On solutions to multivariate maximum-entropy problems," Lecture Notes in Computer Science, v2683, Springer-Verlag, Berlin, pp. 211-228, 2003.

[7]

Cover, T. M. and Thomas, J. A., Elements of Information Theory, Wiley, New York, 1991.

Digital Library

[8]

Cozzolino, J. M. and Zahner, M. J., "The maximum-entropy distribution of the future market price of a stock," Oper. Res., v21, pp. 1200-1211, 1973.

Digital Library

[9]

Dana, R.-A., "A representation result for concave Schur-concave functions," Math. Finance, v15, pp. 613-634, 2005.

[10]

Follmer, H. and Schied, A., Stochastic Finance: An Introduction in Discrete Time, de Gruyter, Berlin, 2004.

[11]

Friedman, C., Huang, J. and Sandow, S., "A utility-based approach to some information measures," Entropy, v9, pp. 1-6, 2007.

[12]

Jaynes, E. T., "Prior probabilities," IEEE Trans. Systems Sci. Cybernetics, v4, pp. 227-251, 1968.

[13]

Jaynes, E. T., "Information theory and statistical mechanics," Physical Rev., v106, pp. 620-630, 1957.

[14]

Johnson, O. and Vignat, C., "Some results concerning maximum Rényi entropy distributions," Annales de l'Institute Henri Poincare (B) Probab. Statist., v43, pp. 339-351, 2007.

[15]

Jouini, E., Schachermayer, W. and Touzi, N., "Law invariant risk measures have the Fatou property," Adv. Math. Econom., v9, pp. 49-71, 2006.

[16]

Kusuoka, S., "On law invariant coherent risk measures," Adv. Math. Econom., v3, pp. 83-95, 2001.

[17]

Lutwak, E., Yang, D. and Zhang, G., "Cramer-Rao and moment-entropy inequalities for Rényi entropy and generalized Fisher information," IEEE Trans. Inform. Theory, v51, pp. 473-478, 2005.

Digital Library

[18]

Markowitz, H. M., "Portfolio selection," J. Finance, v7, pp. 77-91, 1952.

[19]

Rényi, A., "On measures of information and entropy," Proc. 4th Berkeley Sympos. Math. Statist. Probab., University of California Press, Berkeley, pp. 547-561, 1961.

[20]

Rockafellar, R. T. and Uryasev, S., "Optimization of conditional value-at-risk," J. Risk, v2, pp. 21-41, 2000.

[21]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Deviation measures in risk analysis and optimization," 2002.

[22]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Generalized deviations in risk analysis," Finance Stochastics, v10, pp. 51-74, 2006.

[23]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Optimality conditions in portfolio analysis with general deviation measures," Math. Programming, v108, pp. 515-540, 2006a.

Digital Library

[24]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Master funds in portfolio analysis with general deviation measures," J. Banking Finance, v30, pp. 743-777, 2006b.

[25]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Equilibrium with investors using a diversity of deviation measures," J. Banking Finance, v31, pp. 3251-3268, 2007.

[26]

Rockafellar, R. T., Uryasev, S. and Zabarankin, M., "Risk tuning with generalized linear regression," Math. Oper. Res., v33, pp. 712-729, 2008.

Digital Library

[27]

Rudloff, B., "Hedging in incomplete markets and testing compound hypotheses via convex duality," 2006.

[28]

Shannon, C. E., "A mathematical theory of communication," Bell System Tech. J., v27, pp. 379-423, 1948.

[29]

Stutzer, M., "A simple nonparametric approach to derivative security valuation," J. Finance, v51, pp. 1633-1652, 1996.

[30]

Thomas, M. U., "A generalized maximum entropy principle," Oper. Res., v27, pp. 1188-1196, 1979.

Digital Library

[31]

Willard, S., General Topology, Courier Dover Publications, Mineola, NY, 2004.

Cited By

Luo YLiu GPoupart PPan YOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)An alternative to varianceProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668784(60922-60946)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668784
Wang RWei YWillmot G(2020)Characterization, Robustness, and Aggregation of Signed Choquet IntegralsMathematics of Operations Research10.1287/moor.2019.102045:3(993-1015)Online publication date: 1-Aug-2020
https://dl.acm.org/doi/10.1287/moor.2019.1020
Grechuk BZabarankin M(2019)Regression analysisMathematical Programming: Series A and B10.1007/s10107-018-1256-6174:1-2(145-166)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s10107-018-1256-6
Show More Cited By

Recommendations

Some properties of Rényi entropy and Rényi entropy rate

In this paper, we define the conditional Renyi entropy and show that the so-called chain rule holds for the Renyi entropy. Then, we introduce a relation for the rate of Renyi entropy and use it to derive the rate of the Renyi entropy for an irreducible-...
Entropy of fractal systems

The Kolmogorov entropy is an important measure which describes the degree of chaoticity of systems. It gives the average rate of information loss about a position of the phase point on the attractor. Numerically, the Kolmogorov entropy can be estimated ...
On some entropy functionals derived from Rényi information divergence

We consider the maximum entropy problems associated with Renyi Q-entropy, subject to two kinds of constraints on expected values. The constraints considered are a constraint on the standard expectation, and a constraint on the generalized expectation as ...

Comments

Please enable JavaScript to view thecomments powered by Disqus.

Information & Contributors

Information

Published In

cover image Mathematics of Operations Research

Mathematics of Operations Research Volume 34, Issue 2

May 2009

256 pages

ISSN:0364-765X

Issue’s Table of Contents

Publisher

INFORMS

Linthicum, MD, United States

Publication History

Published: 01 May 2009

Received: 19 June 2008

Author Tags

Qualifiers

Article

Contributors

Other Metrics

View Article Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

4
Total Citations
View Citations
0
Total Downloads

Downloads (Last 12 months)0
Downloads (Last 6 weeks)0

Reflects downloads up to 27 Nov 2024

Other Metrics

View Author Metrics

Citations

Cited By

Luo YLiu GPoupart PPan YOh ANaumann TGloberson ASaenko KHardt MLevine S(2023)An alternative to varianceProceedings of the 37th International Conference on Neural Information Processing Systems10.5555/3666122.3668784(60922-60946)Online publication date: 10-Dec-2023
https://dl.acm.org/doi/10.5555/3666122.3668784
Wang RWei YWillmot G(2020)Characterization, Robustness, and Aggregation of Signed Choquet IntegralsMathematics of Operations Research10.1287/moor.2019.102045:3(993-1015)Online publication date: 1-Aug-2020
https://dl.acm.org/doi/10.1287/moor.2019.1020
Grechuk BZabarankin M(2019)Regression analysisMathematical Programming: Series A and B10.1007/s10107-018-1256-6174:1-2(145-166)Online publication date: 1-Mar-2019
https://dl.acm.org/doi/10.1007/s10107-018-1256-6
Wang YWang JLiao HChen H(2017)An efficient semi-supervised representatives feature selection algorithm based on information theoryPattern Recognition10.1016/j.patcog.2016.08.01161:C(511-523)Online publication date: 1-Jan-2017
https://dl.acm.org/doi/10.1016/j.patcog.2016.08.011

View Options

View options

Login options

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

Get this Publication

Media

Figures

Other

Tables

View Issue’s Table of Contents