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Reliable Computing

, Volume 13, Issue 3, pp 261–282 | Cite as

Unimodality, Independence Lead to NP-Hardness of Interval Probability Problems

  • Daniel J. Berleant
  • Olga Kosheleva
  • Vladik Kreinovich
  • Hung T. Nguyen
Article

Abstract

In many real-life situations, we only have partial information about probabilities. This information is usually described by bounds on moments, on probabilities of certain events, etc. –i.e., by characteristics c(p) which are linear in terms of the unknown probabilities p j. If we know interval bounds on some such characteristics \( \underline{a}_i\leq c_i(p)\leq \bar{a}_i \), and we are interested in a characteristic c(p), then we can find the bounds on c(p) by solving a linear programming problem.

In some situations, we also have additional conditions on the probability distribution –e.g., we may know that the two variables x 1 and x 2 are independent, or that the joint distribution of x 1 and x 2 is unimodal. We show that adding each of these conditions makes the corresponding interval probability problem NP-hard.

Keywords

Linear Programming Problem Linear Constraint Stochastic Dominance Unimodal Distribution Integer Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Berleant, D., Cheong, M.-P., Chu, C., Guan, Y., Kamal, A., Sheblé, G., Ferson, S., and Peters, J. F.: Dependable Handling of Uncertainty, Reliable Computing 9 (6) (2003), pp. 407–418.MATHCrossRefGoogle Scholar
  2. 2.
    Berleant, D., Dancre, M., Argaud, J., and Sheblé, G.: Electric Company Portfolio Optimization under Interval Stochastic Dominance Constrraints, in: Cozman, F. G., Nau, R., and Seidenfeld, T. (eds), Proceedings of the 4th International Symposium on Imprecise Probabilities and Their Applications ISIPTA'05, Pittsburgh, Pennsylvania, July 20–24, 2005, pp. 51–57.Google Scholar
  3. 3.
    Berleant, D. J., Kosheleva, O., and Nguyen, H. T.: Adding Unimodality or Independence Makes Interval Probability Problems NP-Hard, in: Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge-Based Systems IPMU'06, Paris, France, July 2–7, 2006 (to appear).Google Scholar
  4. 4.
    Berleant, D., Xie, L., and Zhang, J.: Statool: A Tool for Distribution Envelope Determination (DEnv), an Interval-Based Algorithm for Arithmetic on Random Variables, Reliable Computing 9 (2) (2003), pp. 91–108.MATHCrossRefGoogle Scholar
  5. 5.
    Berleant, D. and Zhang, J.: Representation and Problem Solving with the Distribution Envelope Determination (DEnv) Method, Reliability Engineering and System Safety 85 (1–3) (2004).Google Scholar
  6. 6.
    Berleant, D. and Zhang, J.: Using Pearson Correlation to Improve Envelopes around the Distributions of Functions, Reliable Computing 10 (2) (2004), pp. 139–161.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Borglin, A. and Keiding, H.: Stochastic Dominance and Conditional Expectation—An Insurance Theoretical Approach, The Geneva Papers on Risk and Insurance Theory 27 (2002), pp. 31–48.CrossRefGoogle Scholar
  8. 8.
    Chateauneuf, A.: On the Use of Capacities in Modeling Uncertainty Aversion and Risk Aversion, Journal of Mathematical Economics 20 (1991), pp. 343–369.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Ferson, S.: RAMAS Risk Calc 4.0, CRC Press, Boca Raton, Florida.Google Scholar
  10. 10.
    Garey, M. R. and Johnson, D. S.: Computers and Intractability, a Guide to the Theory of NP-Completeness, W. H. Freemand and Company, San Francisco, 1979.MATHGoogle Scholar
  11. 11.
    Horowitz, J., Manski, C. F., Ponomareva, M., and Stoye, J.: Computation of Bounds on Population Parameters When the Data Are Incomplete, Reliable Computing 9 (6) (2003), pp. 419–440.MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Klir, G. J., Xiang, G., and Kosheleva, O.: Estimating Information Amount under Interval Uncertainty: Algorithmic Solvability and Computational Complexity, in: Proceedings of the International Conference on Information Processing and Management of Uncertainty in Knowledge- Based Systems IPMU'06, Paris, France, July 2–7, 2006 (to appear).Google Scholar
  13. 13.
    Kreinovich, V. and Ferson, S.: Computing Best-Possible Bounds for the Distribution of a Sum of Several Variables is NP-Hard, International Journal of Approximate Reasoning 41 (2006), pp. 331–342.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Kreinovich, V., Lakeyev, A., Rohn, J., and Kahl, P.: Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, 1997.Google Scholar
  15. 15.
    Kreinovich, V., Pauwels, E. J., Ferson, S., and Ginzburg, L.: A Feasible Algorithm for Locating Concave and Convex Zones of Interval Data and Its Use in Statistics-Based Clustering, Numerical Algorithms 37 (2004), pp. 225–232.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Krymsky, V. G.: Computing Interval Estimates for Components of Statistical Information with Respect to Judgements on Probability Density Functions, in: Dongarra, J., Madsen, K., and Wasniewski, J. (eds), PARA'04 Workshop on State-of-the-Art in Scientific Computing, Springer Lecture Notes in Computer Science 3732, 2005, pp. 151–160.Google Scholar
  17. 17.
    Kuznetsov, V. P.: Auxiliary Problems of Statistical Data Processing: Interval Approach, in: Proceedings of the International Workshop on Applications of Interval Computations APIC'95, El Paso, Texas, February 23–25, 1995 (a special supplement to the journal Reliable Computing), pp. 123–129.Google Scholar
  18. 18.
    Kuznetsov, V. P.: Interval Methods for Processing Statistical Characteristics, in: Proceedings of the International Workshop on Applications of Interval Computations APIC'95, El Paso, Texas, February 23–25, 1995 (a special supplement to the journal Reliable Computing), pp. 116–122.Google Scholar
  19. 19.
    Kuznetsov, V.: Interval Statistical Models, Radio i Svyaz, Moscow, 1991 (in Russian).Google Scholar
  20. 20.
    Manski, C. F.: Partial Identification of Probability Distributions, Springer Verlag, New York, 2003.MATHGoogle Scholar
  21. 21.
    Sivaganesan, S. and Berger, J. O.: Ranges of Posterior Measures for Priors with Unimodal Contaminations, Annals of Statistics 17 (2) (1989), pp. 868–889.MATHMathSciNetGoogle Scholar
  22. 22.
    Skulj, D.: Generalized Conditioning in Neighbourhood Models, in: Cozman, F. G., Nau, R., and Seidenfeld, T. (eds), Proceedings of the 4th International Symposium on Imprecise Probabilities and Their Applications ISIPTA'05, Pittsburgh, Pennsylvania, July 20–24, 2005.Google Scholar
  23. 23.
    Skulj, D.: A Role of Jeffrey's Rule of Conditioning in Neighborhood Models, to appear.Google Scholar
  24. 24.
    Villaverde, K. and Kreinovich, V.: A Linear-Time Algorithm That Locates Local Extrema of a Function of One Variable from Interval Measurement Results, Interval Computations 4 (1993), pp. 176–194.MathSciNetGoogle Scholar
  25. 25.
    Villaverde, K. and Kreinovich, V.: Parallel Algorithm That Locates Local Extrema of a Function of One Variable from Interval Measurement Results, Reliable Computing, 1995, Supplement (Extended Abstracts of APIC'95: International Workshop on Applications of Interval Computations, El Paso, TX, Febr. 23–25, 1995), pp. 212–219.Google Scholar
  26. 26.
    Walley, P.: Statistical Reasoning with Imprecise Probabilities, Chapman & Hall, N.Y., 1991.MATHGoogle Scholar
  27. 27.
    Zhang, J. and Berleant, D.: Arithmetic on Random Variables: Squeezing the Envelopes with New Joint Distribution Constraints, in: Proceedings of the 4th International Symposium on Imprecise Probabilities and Their Applications ISIPTA'05, Pittsburgh, Pennsylvania, July 20–24, 2005, pp. 416–422.Google Scholar

Copyright information

© Springer Science + Business Media B.V. 2007

Authors and Affiliations

  • Daniel J. Berleant
    • 1
  • Olga Kosheleva
    • 2
  • Vladik Kreinovich
    • 2
  • Hung T. Nguyen
    • 3
  1. 1.Department of Information ScienceUniversity of Arkansas at Little RockLittle RockUSA
  2. 2.NASA Pan-American Center for Earth and Environmental Studies (PACES)University of TexasEl PasoUSA
  3. 3.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

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