Abstract
We study scaled Bregman distances between distributions from exponential families, respectively, data-derived empirical distributions (relative frequencies, histograms). For the scaling, we also employ distribution mixtures. The outcoming parameter-dependences constitute (random) surfaces which offer a basis for computer-graphical exploratory analyses about the internal structure of exponential families, as well as for concrete 3D computer-graphical statistical decision making such as simultaneous parameter estimation and goodness-of-fit investigations. Morever, we study the distributional asymptotics of random scaled Bregman distances where the sample size of the involved empirical distribution tends to infinity. Small-sample-size results and a comparison with the prominent quantile-quantile-plot technique will be shown, too. ...
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References
Amari, S.-I.: Integration of stochastic models by minimizing α-divergence. Neural Computation 19(10), 2780–2796 (2007)
Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J. Machine Learning Research 6, 1705–1749 (2005)
Bartlett, P.L., Jordan, M.I., McAuliffe, J.D.: Convexity, classification and risk bounds. JASA 101, 138–156 (2006)
Boratynska, A.: Stability of Bayesian inference in exponential families. Statist. & Probab. Letters 36, 173–178 (1997)
Carlson, B.A., Clements, M.A.: A computationally compact divergence measure for speech processing. IEEE Transactions on PAMI 13, 1255–1260 (1991)
Censor, Y., Zenios, S.A.: Parallel Optimization - Theory, Algorithms, and Applications. Oxford University Press, New York (1997)
Cesa-Bianchi, N., Lugosi, G.: Prediction, Learning, Games. Cambridge University Press, Cambridge (2006)
Do, M.N., Vetterli, M.: Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance. IEEE Transactions on Image Processing 11, 146–158 (2002)
Freund, Y., Schapire, R.E.: A decision-theoretic generalization of on-line learning and an application to boosting. J. Comput. Syst. Sci. 55, 119–139 (1997)
Hertz, T., Bar-Hillel, A., Weinshall, D.: Learning distance functions for information retrieval. In: Proc. IEEE Comput. Soc. Conf. on Computer Vision and Pattern Rec. CVPR, vol. 2, pp. II-570–II-577 (2004)
Lafferty, J.D.: Additive models, boosting, and inference for generalized divergences. In: Proceedings of the Twelfth Annual Conference on Computational Learning Theory, pp. 125–133. ACM Press, New York (1999)
Marquina, A., Osher, S.J.: Image super-resolution by TV-regularization and Bregman iteration. J. Sci. Comput. 37, 367–382 (2008)
Nock, R., Nielsen, F.: Bregman divergences and surrogates for learning. IEEE Transactions on PAMI 31(11), 2048–2059 (2009)
Pardo, M.C., Vajda, I.: On asymptotic properties of information-theoretic divergences. IEEE Transaction on Information Theory 49(7), 1860–1868 (2003)
Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging. Springer, New York (2008)
Stummer, W.: Some Bregman distances between financial diffusion processes. Proc. Appl. Math. Mech. 7(1), 1050503–1050504 (2007)
Stummer, W., Vajda, I.: On Bregman Distances and Divergences of Probability Measures. IEEE Transaction on Information Theory 58(3), 1277–1288 (2012)
Teboulle, M.: A unified continuous optimization framework for center-based clustering methods. Journal of Machine Learning Research 8, 65–102 (2007)
Vajda, I., Zvárová, J.: On generalized entropies, Bayesian decisions and statistical diversity. Kybernetika 43(5), 675–696 (2007)
Veldhuis, R.N.J.: The centroid of the Kullback-Leibler distance. IEEE Signal Processing Letters 9(3), 96–99 (2002)
Xu, J., Osher, S.: Iterative regularization and nonlinear inverse scale space applied to wavelet-based denoising. IEEE Transaction on Image Processing 16(2), 534–544 (2007)
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Kißlinger, AL., Stummer, W. (2013). Some Decision Procedures Based on Scaled Bregman Distance Surfaces. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2013. Lecture Notes in Computer Science, vol 8085. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40020-9_52
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DOI: https://doi.org/10.1007/978-3-642-40020-9_52
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