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. ...
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
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)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-642-40020-9_52
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-40019-3
Online ISBN: 978-3-642-40020-9
eBook Packages: Computer ScienceComputer Science (R0)