Equivocation Axiom on First Order Languages
In this paper we investigate some mathematical consequences of the Equivocation Principle, and the Maximum Entropy models arising from that, for first order languages. We study the existence of Maximum Entropy models for these theories in terms of the quantifier complexity of the theory and will investigate some invariance and structural properties of such models.
KeywordsMaximum Entropy models Probabilistic reasoning Equivocation principle
- 1.Bacchus, F., A. J. Grove, J. Y. Halpern, and D. Koller, Generating new beliefs from old, Proceedings of the Tenth Annual Conference on Uncertainty in Artificial Intelligence, (UAI-94), 1994, pp. 37–45.Google Scholar
- 3.Berger A., Della Pietra S., Della Pietra V.: A maximum entropy approach to natural language processing. Computational Linguistics 22(1), 39–71 (1996)Google Scholar
- 4.Chen, C. H., Maximum entropy analysis for pattern recognition, in P. F. Fougere (ed.), Maximum Entropy and Bayesian Methods, Kluwer Academic Publisher, London, 1990.Google Scholar
- 6.Grotenhuis, M. G., An Overview of the Maximum Entropy Method of Image Deconvolution, A University of Minnesota Twin Cities Plan B Masters paper.Google Scholar
- 7.Grove A. J., J. Y. Halpern, and D. Koller, Asymptotic conditional probabilities: the unary case, SIAM Journal of Computing 25(1):1–51, 1996.Google Scholar
- 8.Jaynes, E. T., Information theory and statistical mechanics, Physical Reviews 106:620–630, 108:171–190, 1957.Google Scholar
- 9.Jaynes, E. T., Notes on present status and future prospects, in W. T. Grandy and L. H. Schick (eds.), Maximum Entropy and Bayesian Methods, Kluwer, London, 1990, pp. 1–13.Google Scholar
- 10.Jaynes, E. T., How Should We Use Entropy in Economics? 1991, manuscript available at: http://www.leibniz.imag.fr/LAPLACE/Jaynes/prob.html.
- 11.Kapur J. N.: Twenty five years of maximum entropy. Journal of Mathematical and Physical Sciences 17(2), 103–156 (1983)Google Scholar
- 12.Kapur J. N.: Non-additive measures of entropy and distributions of statistical mechanics. Indian Journal of Pure and Applied Mathematics 14(11), 1372–1384 (1983)Google Scholar
- 15.Paris J. B.: The Uncertain Reasoner’s Companion. Cambridge University Press, Cambridge (1994)Google Scholar
- 18.Paris, J. B., and S. Rafiee Rad, A note on the least informative model of a theory, in F. Ferreira, B. Löwe, E. Mayordomo, and L. Mendes Gomes (eds.), Programs Proofs Processes, CiE 2010, Springer LNCS 6158, 2010, pp. 342–351.Google Scholar
- 19.Rafiee Rad, S., Inference Processes For Probabilistic First Order Languages. PhD Thesis, University of Manchester, 2009. http://www.maths.manchester.ac.uk/~jeff/
- 20.Rosenkrantz, R. D., Inference, Method and Decision: Towards a Bayesian Philosophy of Science, Reidel, Dordrecht, 1977.Google Scholar
- 21.Shannon C. E., Weaver W.: The Mathematical Theory of Communication. University of Illinois Press, Champaign (1949)Google Scholar
- 22.Williamson J.: Bayesian nets and causality: philosophical and computational foundations. Oxford University Press, Oxford (2005)Google Scholar
- 23.Williamson J.: Objective Bayesian probabilistic logic. Journal of Algorithms in Cognition, Informatics and Logic 63, 167–183 (2008)Google Scholar
- 24.Williamson, J., In Defence of Objective Bayesianism, Oxford University Press, Oxford, 2010, pp. 167–183.Google Scholar
- 25.Zellner, A., Bayesian methods and entropy in economics and econometrics, in W. T. Grandy and L. H. Schick (eds.), Maximum Entropy and Bayesian Methods, 1991.Google Scholar
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