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On the Computation of Entropy Prior Complexity and Marginal Prior Distribution for the Bernoulli Model

  • N. Balakrishnan
  • C. Koukouvinos
  • C. Parpoula
Article

Abstract

As the size and complexity of models grow, the choice of the best model becomes a difficult and challenging task. Once the best model is specified, the goodness of fit of the model needs to be examined first. A highly complex model may provide a good fit, but giving no consideration to model complexity could result in incorrect estimates of parameter values and predictions. In order to improve the model selection process, model complexity needs to be defined clearly. This article studies different aspects of model complexity and discusses the extent to which they can be measured. The most common attribute that is usually ignored from many complexity measures is the parameter prior, which is an inherent part of the model and could impact the complexity significantly. The concept of parameter prior and its connection to model complexity are therefore discussed here, and some relationships to the entropy measure elements are also addressed.

Keywords

Model complexity Entropy Prior odds Posterior odds Marginal probability 

AMS Subject Classification

62K15 62-07 62J12 

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References

  1. Balakrishnan, N., C. Koukouvinos, and C. Parpoula. 2012. Analysis of a supersaturated design using entropy prior complexity for binary responses via generalized linear models. Stat. Methodol., 9, 478–185.MathSciNetCrossRefGoogle Scholar
  2. Balasubramanian, V. 1997. Statistical inference, Occam’s Razor, and statistical mechanics on the space of probability distributions. Neural Comput., 9, 349–368.CrossRefGoogle Scholar
  3. Bennett, C. H. 1986. On the nature and origin of complexity in discrete, homogeneous locally-interacting systems. Found. Phys., 16, 585–592.MathSciNetCrossRefGoogle Scholar
  4. Berger, A. L, S. Della Pietra, and V. J. Della Pietra. 1996. A maximum-entropy approach to natural language processing. Comput. Linguistics, 22, 39–71.Google Scholar
  5. Bialek, W., I. Nemenman, and N. Tishby. 2001. Predictability, complexity, and learning. Neural Comput., 13, 2409–2463.CrossRefGoogle Scholar
  6. Brooks, R. J., and A. M. Tobias. 1996. Choosing the best model: Level of detail, complexity and model performance. Math. Comput. Model., 24, 1–14.CrossRefGoogle Scholar
  7. Brookshear, J. G. 1989. Theory of computation: Formal languages, automata, and complexity. Redwood City, CA: Benjamin-Cummings Publishing Company.zbMATHGoogle Scholar
  8. Bueso, M. C., G. Qian, and J. M. Angulo. 1999. Stochastic complexity and model selection from incomplete data. J. Stat. Plan. Inference, 76, 273–284.CrossRefGoogle Scholar
  9. Catalan, R. G., J. Garay, and R. López-Ruiz. 2002. Features of the extension of a statistical measure of complexity for continuous systems. Phys. Rev. E, 66, 011102(6).Google Scholar
  10. Caticha, A. 2007. Information and entropy. In Bayesian inference and maximum entropy methods in science and engineering, ed. K. Knuth et al., AIP Conf. Proc., vol. 954, 11. New York, NY: AIP.Google Scholar
  11. Charles, S. B. 2002. A comparison of marginal likelihood computation methods. In COMPSTAT 2002: Proceedings in computational statistics, ed. W. Härdle and B. Ronz, 111–117. Berlin, Heidelberg: Springer-Verlag.Google Scholar
  12. Crutchfield, J. P., and K. Young. 1989. Inferring statistical complexity. Phys. Rev. Lett., 63, 105–108.MathSciNetCrossRefGoogle Scholar
  13. Della Pietra, S., V. J. Della Pietra, and J. D. Lafferty. 1997. Inducing features of random fields. IEEE Trans. Pattern Anal. Machine Intelligence, 19, 380–393.CrossRefGoogle Scholar
  14. Dunn, J. 2000. Model complexity: The fit to random data reconsidered. Psychol. Res., 63, 174–182.CrossRefGoogle Scholar
  15. Feldman, D. P., and J. P. Crutchfield. 1998. Measures of statistical complexity. Phys. Lett. A, 238, 244–252.MathSciNetCrossRefGoogle Scholar
  16. Grünwald, P. D. 2005. MDL tutorial. In Advances in minimum description length: Theory and applications, ed. P. D. Grünwald, I. J. Myung, and M. A. Pitt, 16–17. Cambridge, MA: MIT Press.Google Scholar
  17. Grünwald, P. D. 2007. The minimum description length principle. Cambridge, MA: MIT Press.Google Scholar
  18. Hall, P., and J. Hannan. 1988. On stochastic complexity and nonparametric density estimation. Biometrika, 75, 705–714.MathSciNetCrossRefGoogle Scholar
  19. Hansen, A. J., and B. Yu. 2001. Model selection and the principle of minimum description length. J. Am. Stat. Assoc., 96, 746–774.MathSciNetCrossRefGoogle Scholar
  20. Hopcroft, J. E., R. Motwani, and J. D. Ullman. 2000. Introduction to automata theory, languages, and computation, 3rd ed. Reading, MA: Addison-Wesley.zbMATHGoogle Scholar
  21. Jaynes, E. T. 2003. Probability theory—The logic of science. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  22. Kass, R. E., and A. E. Raftery. 1995. Bayes factors. J. Am. Stat. Assoc., 90, 773–795.MathSciNetCrossRefGoogle Scholar
  23. Lee, M. D. 2002. Generating additive clustering models with minimal stochastic complexity. J. Classification, 19, 69–85.MathSciNetCrossRefGoogle Scholar
  24. Li, M., and P. M. B. Vitanyi. 1993. An introduction to Kolmogorov complexity and its applications. New York, NY: Springer-Verlag.CrossRefGoogle Scholar
  25. López-Ruiz, R., H. L. Mancini, and X. Calbet. 1995. A statistical measure of complexity. Phys. Lett. A, 209, 321–326.CrossRefGoogle Scholar
  26. Myung, I. J., and M. A. Pitt. 1997. Applying Occam’s razor in modeling cognition: A Bayesian approach. Psychonomic Bull. Rev., 4, 79–95.CrossRefGoogle Scholar
  27. Myung, I. J. 2000. The importance of complexity in model selection. J. Math. Psychol., 44, 190–204.CrossRefGoogle Scholar
  28. Myung, I. J., V. Balasubramanian, and M. A. Pitt. 2000. Counting probability distributions: Differential geometry and model selection. Proc. Nat. Acad. Sci. USA, 97, 11170–11175.MathSciNetCrossRefGoogle Scholar
  29. Rissanen, J. 1986. Stochastic complexity and modeling. Ann. Statistics, 14, 1080–1100.MathSciNetCrossRefGoogle Scholar
  30. Rissanen, J. 1987. Stochastic complexity (with discussion). J. R. Stat. Soc. Ser. B, 49, 223–265.MathSciNetzbMATHGoogle Scholar
  31. Rissanen, J. 1989. Stochastic complexity in statistical inquiry. Singapore: World Scientific Publishing Company.zbMATHGoogle Scholar
  32. Rissanen, J. 1996. Fisher information and stochastic complexity. IEEE Trans. Information Theory, 42, 40–47.MathSciNetCrossRefGoogle Scholar
  33. Rissanen, J. 2005. Complexity and information in modeling. Chapter IV In Computability, complexity and constructivity in economic analysis, ed. K. Velupillai, chap. IV. Oxford, UK: Blackwell.Google Scholar
  34. Rissanen, J. 2007. Information and complexity in statistical modeling. New York, NY: Springer-Verlag.zbMATHGoogle Scholar
  35. Rissanen, J. 2012. Optimal estimation of parameters. Cambridge, UK: Cambridge University Press.CrossRefGoogle Scholar
  36. Shannon, C. E. 1948. A mathematical theory of communication. Bell System Tech. J., 27, 379–423, 623–656.MathSciNetCrossRefGoogle Scholar
  37. Spiegelhalter, D. J., N. G. Best, B. P. Carlin, and A. Van der Linde. 2002. Bayesian measures of model complexity and fit. J. R. Stat. Soc. Ser. B, 64, 583–639 (with discussion).MathSciNetCrossRefGoogle Scholar
  38. Van der Linde, A. 2012. A Bayesian view of model complexity. Stat. Neerland., 66, 253–271.MathSciNetCrossRefGoogle Scholar
  39. Vanpaemel, W. 2009. Measuring model complexity with the prior predictive. In Advances in neural information processing systems (NIPS), ed. Y. Bengio, D. Schuurmans, J. Lafferty, C. K. I. Williams, and A. Culotta, vol. 22, 1919–1927. Red Hook, NY: Curran Associates.Google Scholar
  40. Wallis, K. F. 2006. A note on the calculation of entropy from histograms. Unpublished paper, University of Warwick, Coventry, UK.Google Scholar

Copyright information

© Grace Scientific Publishing 2015

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcMaster UniversityHamiltonCanada
  2. 2.Department of StatisticsKing Abdulaziz UniversityJeddahSaudi Arabia
  3. 3.Department of MathematicsNational Technical University of AthensZografou, AthensGreece

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