Abstract
A framework for comparing the maximum likelihood (ML) and maximum entropy (ME) approaches is developed. Two types of linear models are considered. In the first type, the objective is to estimate probability distributions given some moment conditions. In this case the ME and ML are equivalent. A generalization of this type of estimation models to incorporate noisy data is discussed as well. The second type of models encompasses the traditional linear regression type models where the number of observations is larger than the number of unknowns and the objects to be inferred are not natural probabilities. After reviewing a generalized ME estimator and the empirical likelihood (or weighted least squares) estimator, the two are compared and contrasted with the ML. It is shown that, in general, the ME estimators use less input information and may be viewed, within the second type models, as expected log-likelihood estimators. In terms of informational ranking, if the objective is to estimate with minimum a-priori assumptions, then the generalized ME estimator is superior to the other estimators. Two detailed examples, reflecting the two types of models, are discussed. The first example deals with estimating a first order Markov process. In the second example the empirical (natural) weights of each observation, together with the other unknowns, are the subject of interest.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Akaike, H (1986), “The Selection Smoothness Priors for Distributed Lag Estimation,” in Bayesian and Decision Techniques: Essays in Honor of Bruno de Finetti, eds. P. K. Goel and A. Zellner (Amsterdam, North-Holland) 109–118.
Agmon, N., Y. Alhassid, and R. D. Levine (1979), “An Algorithm for Finding the Distribution of Maximal Entropy,” Journal of Computational Physics, Vol. 30, pp. 250–259.
Bernardo, J.M. (1979), “Expected Information as Expected Utility,” The Annals of Statistics, 7,686–690.
Csiszar, I. (1991) “Why Least Squares and Maximum Entropy? An Axiomatic Approach to Inference for Linear Inverse Problems,” The Annals of Statistics, 19, 2032–2066.
DiCiccio, T., P. Hall, and J. Romano (1991), “Empirical Likelihood is Bartlett-Correctable,” The Annals of Statistics, 19, 1053–1061.
Golan, A., and G. G. Judge and D. Miller (1996), Maximum Entropy Econometrics: Robust Estimation With Limited Data, John Wiley & Sons, New York.
Golan, A., and G. Judge (1996), “A Maximum Entropy Approach to Empirical Likelihood Estimation and Inference,” (Working paper), UC Berkeley.
Golan, A., and S. J. Vogel (1997), “Estimation of Stationary and Non-Stationary Social Accounting Matrix Coefficients With Structural and Supply-Side Information,” (Working paper).
Good, I. J. (1963), “Maximum Entropy for Hypothesis Formulation, Especially for Multidimensional Contingency Tables,” Annals of Mathematical Statistics, Vol. 34, pp. 911–934.
Hall, P. (1990), “Pseudo-Likelihood Theory for Empirical Likelihood,” The Annals of Statistics, 18, 121–140.
Imbens, G.W. (1993), “A New Approach to Generalized Method of Moments Estimation,” (mimeo), Harvard University.
Imbens, G.W. and J.K. Hellerstein (1994), “Imposing Moment Restrictions by Weighting,” (mimeo), Harvard University.
Jaynes, E.T. (1957a), “Information Theory and Statistical Mechanics,” Physics Review, 106, 620–630.
Jaynes, E.T. (1957b), “Information Theory and Statistical Mechanics II,” Physics Review, 108, 171–190.
Jaynes, E.T. (1963), “Information Theory and Statistical Mechanics II,” in K.W. Ford (ed.), Statistical Physics, W.A. Benamin, Inc., New York, 181–218.
Jaynes, E.T. (1984), “Prior Information and Ambiguity in Inverse Problems,” in D.W. McLaughlin (ed.), Inverse Problems, SIAM Proceedings, American Mathematical Society, Providence, RI, 151–166.
Kullback, J. (1959), Information Theory and Statistics, New York: John Wiley & Sons.
Levine, R.D. (1980), “An Information Theoretical Approach to Inversion Problems,” Journal of Physics, A, 13, 91–108.
McFadden, D. (1974), “The Measurement of Urban Travel demand,” Journal of Public Economics, Vol. 3, pp. 303–328.
Owen, A. (1990), “Empirical Likelihood Ratio Confidence Regions,” The Annals of Statistics, 18, 90–120.
Owen, A. (1991), “Empirical Likelihood for Linear Models,” The Annals of Statistics, 19, 1725–1747.
Qin, J. and J. Lawless (1994), “Empirical Likelihood and General Estimating Equations,” The Annals of Statistics, 22, 300–325.
Shannon, C.E. (1948), “A Mathematical Theory of Communication,” Bell System Technical Journal, 27, 379–423.
Shore, J.E. and R.W. Johnson (1980), “Axiomatic Derivation of the Principle of Maximum Entropy and the Principle of Minimum Cross-Entropy,” IEEE Transactions on Information Theory, IT-26(1), 26–37.
Skilling, J. (1989), “The Axioms of Maximum Entropy,” in J. Skilling (ed.), Maximum Entropy and Bayesian Methods in Science and Engineering, Kluwer Academic, Dordrecht, 173–187.
Tobias, J., and A. Zellner (1997), Further Results on Bayesian Method of Moments Analysis of the Multiple Regression Model. H.G.B. Alexander Research Foundation, University of Chicago.
Zellner, A. (1988), “Optimal Information Processing and Bayes Theorem,” American Statistician, 42, 278–284.
A. Zellner. Bayesian method of moments/ instrumental variable (BMOM/iv) analysis of mean and regression models. In J.C. Lee, W.C. Johnson, and A. Zellner, editors, Modeling and Prediction: Honoring Seymour Geisser, pages 61–75. Springer-Verlag, 1996.
A. Zellner (1997). The Bayesian method of moments (BMOM): theory and applications. In T. Fomby and R.C. Hill, editors, Advances in Econometrics.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media Dordrecht
About this paper
Cite this paper
Golan, A. (1998). Maximum Entropy, Likelihood and Uncertainty: A Comparison. In: Erickson, G.J., Rychert, J.T., Smith, C.R. (eds) Maximum Entropy and Bayesian Methods. Fundamental Theories of Physics, vol 98. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-5028-6_4
Download citation
DOI: https://doi.org/10.1007/978-94-011-5028-6_4
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-6111-7
Online ISBN: 978-94-011-5028-6
eBook Packages: Springer Book Archive