Scaling of model approximation errors and expected entropy distances. (English) Zbl 1517.62059

Summary: We compute the expected value of the Kullback-Leibler divergence of various fundamental statistical models with respect to Dirichlet priors. For the uniform prior, the expected divergence of any model containing the uniform distribution is bounded by a constant \(1-\gamma\). For the models that we consider this bound is approached as the cardinality of the sample space tends to infinity, if the model dimension remains relatively small. For Dirichlet priors with reasonable concentration parameters the expected values of the divergence behave in a similar way. These results serve as a reference to rank the approximation capabilities of other statistical models.


62F15 Bayesian inference
62B10 Statistical aspects of information-theoretic topics
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[1] Ay, N.: An information-geometric approach to a theory of pragmatic structuring. Ann. Probab. 30 (2002), 416-436. · Zbl 1010.62007 · doi:10.1214/aop/1020107773
[2] Drton, M., Sturmfels, B., Sullivant, S.: Lectures on Algebraic Statistics. Birkhäuser, Basel 2009. · Zbl 1166.13001
[3] Frigyik, B. A., Kapila, A., Gupta, M. R.: Introduction to the Dirichlet Distribution and Related Processes. Technical Report, Department of Electrical Engineering University of Washington, 2010.
[4] Matúš, F., Ay, N.: On maximization of the information divergence from an exponential family. Proc. WUPES’03, University of Economics, Prague 2003, pp. 199-204.
[5] Matúš, F., Rauh, J.: Maximization of the information divergence from an exponential family and criticality. Proc. ISIT, St. Petersburg 2011, pp. 903-907.
[6] Montúfar, G., Rauh, J., Ay, N.: Expressive power and approximation errors of restricted Boltzmann machines. Advances in NIPS 24, MIT Press, Cambridge 2011, pp. 415-423.
[7] Nemenman, I., Shafee, F., Bialek, W.: Entropy and inference, revisited. Advances in NIPS 14, MIT Press, Cambridge 2001, pp. 471-478.
[8] Rauh, J.: Finding the Maximizers of the Information Divergence from an Exponential Family. Ph.D. Thesis, Universität Leipzig 2011. · Zbl 1365.94160
[9] Rauh, J.: Optimally approximating exponential families. Kybernetika 49 (2013), 199-215. · Zbl 1283.94027
[10] Wolpert, D., Wolf, D.: Estimating functions of probability distributions from a finite set of samples. Phys, Rev. E 52 (1995), 6841-6854. · doi:10.1103/PhysRevE.52.6841
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