Symmetric measures via moments

Algebraic tools in statistics have recently been receiving special attention and a number of interactions between algebraic geometry and computational statistics have been rapidly developing. This paper presents another such connection, namely, one between probabilistic models invariant under a finite group of (non-singular) linear transformations and polynomials invariant under the same group. Two specific aspects of the connection are discussed: generalization of the (uniqueness part of the multivariate) problem of moments and log-linear, or toric, modeling by expansion of invariant terms. A distribution of minuscule subimages extracted from a large database of natural images is analyzed to illustrate the above concepts.

[1]  Solomon Kullback,et al.  Information Theory and Statistics , 1960 .

[2]  H. O. Foulkes.  Abstract Algebra , 1967, Nature.

[3]  I. Csiszár.  $I$-Divergence Geometry of Probability Distributions and Minimization Problems , 1975 .

[4]  Rodney W. Johnson,et al.  Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy , 1980, IEEE Trans. Inf. Theory.

[5]  K. Mardia,et al.  A Note on the Maximum-Entropy Principle , 1983 .

[6]  L. Mead,et al.  Maximum entropy in the problem of moments , 1984 .

[7]  A. Lippman.  A Maximum Entropy Method for Expert System Construction , 1988 .

[8]  R. Dykstra,et al.  Duality of I Projections and Maximum Likelihood Estimation for Log-Linear Models Under Cone Constraints , 1988 .

[9]  R. Durrett.  Probability: Theory and Examples , 1993 .

[10]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[11]  I. Csiszár.  Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems , 1991 .

[12]  A. Barron,et al.  APPROXIMATION OF DENSITY FUNCTIONS BY SEQUENCES OF EXPONENTIAL FAMILIES , 1991 .

[13]  Bernd Sturmfels,et al.  Algorithms in invariant theory , 1993, Texts and monographs in symbolic computation.

[14]  Larry Smith,et al.  Polynomial Invariants of Finite Groups , 1995 .

[15]  M. Schervish.  Theory of Statistics , 1995 .

[16]  Eberhard Zeidler,et al.  Applied Functional Analysis: Applications to Mathematical Physics , 1995 .

[17]  F. Y. Edgeworth,et al.  The theory of statistics , 1996 .

[18]  Christian Berg,et al.  Moment problems and polynomial approximation , 1996 .

[19]  Amos Golan,et al.  Maximum entropy econometrics , 1996 .

[20]  Song-Chun Zhu,et al.  Prior Learning and Gibbs Reaction-Diffusion , 1997, IEEE Trans. Pattern Anal. Mach. Intell..

[21]  John D. Lafferty,et al.  Inducing Features of Random Fields , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[22]  David A. Cox,et al.  Ideals, Varieties, and Algorithms , 1997 .

[23]  Song-Chun Zhu,et al.  Minimax Entropy Principle and Its Application to Texture Modeling , 1997, Neural Computation.

[24]  J. V. van Hateren,et al.  Independent component filters of natural images compared with simple cells in primary visual cortex , 1998, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  Song Chun ZhuDept.  Clutter Modeling and Performance Analysis in Automatic Target Recognition 1 , 1998 .

[26]  D. Geman,et al.  Invariant Statistics and Coding of Natural Microimages , 1998 .

[27]  Michael I. Jordan.  Graphical Models , 1998 .

[28]  Song-Chun Zhu,et al.  Embedding Gestalt Laws in Markov Random Fields , 1999, IEEE Trans. Pattern Anal. Mach. Intell..

[29]  David Mumford,et al.  Statistics of natural images and models , 1999, Proceedings. 1999 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (Cat. No PR00149).

[30]  Krein's Strings, the Symmetric Moment Problem, and Extending a Real Positive Definite Function , 1999 .

[31]  Jordan Stoyanov,et al.  Krein condition in probabilistic moment problems , 2000 .

[32]  O. Knill.  On Hausdorff ’ s moment problem in higher dimensions , 2000 .

[33]  Donald Geman,et al.  Modeling natural microimage statistics , 2000 .

[34]  Michael Junk,et al.  MAXIMUM ENTROPY FOR REDUCED MOMENT PROBLEMS , 2000 .

[35]  Marcel de Jeu,et al.  Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights , 2001, math/0111019.

[36]  Flemming Topsøe,et al.  Basic Concepts, Identities and Inequalities - the Toolkit of Information Theory , 2001, Entropy.

[37]  D. Mumford,et al.  Stochastic models for generic images , 2001 .

[38]  Jong-Wuu Wu,et al.  Criteria for the Unique Determination of Probability Distributions by Moments , 2001 .

[39]  Paul A. Viola,et al.  Robust Real-time Object Detection , 2001 .

[40]  Ann B. Lee,et al.  The Complex Statistics of High-Contrast Patches in Natural Images , 2001 .

[41]  J. Fogarty.  On Noether's bound for polynomial invariants of a finite group. , 2001 .

[42]  Kim Steenstrup Pedersen,et al.  Toward a Full Probability Model of Edges in Natural Images , 2002, ECCV.

[43]  Harm Derksen,et al.  Computational Invariant Theory , 2002 .

[44]  J. Besag.  What is a statistical model? Discussion , 2002 .

[45]  Ximing Wu,et al.  Calculation of maximum entropy densities with application to income distribution , 2003 .

[46]  Kim Steenstrup Pedersen,et al.  Statistics of Natural Image Geometry , 2003 .

[47]  Ann B. Lee.  The Nonlinear Statistics of High-Contrast Patches in Natural Images , 2003 .

[48]  Eric R. Ziegel,et al.  The Elements of Statistical Learning , 2003, Technometrics.

[49]  Donald Geman,et al.  Coarse-to-Fine Face Detection , 2004, International Journal of Computer Vision.

[50]  Kim Steenstrup Pedersen,et al.  The Nonlinear Statistics of High-Contrast Patches in Natural Images , 2003, International Journal of Computer Vision.

[51]  Paul A. Viola,et al.  Robust Real-Time Face Detection , 2001, International Journal of Computer Vision.

[52]  Stephen E. Fienberg,et al.  Testing Statistical Hypotheses , 2005 .

[53]  Christoph Schnörr,et al.  Natural Image Statistics for Natural Image Segmentation , 2005, International Journal of Computer Vision.

[54]  Pierre Moulin,et al.  On the existence and characterization of the maxent distribution under general moment inequality constraints , 2005, IEEE Transactions on Information Theory.

[55]  L. Pachter,et al.  Algebraic Statistics for Computational Biology: Preface , 2005 .

[56]  Donald Geman,et al.  Ordinal Coding of Image Microstructure , 2006, IPCV.

[57]  D. Geiger,et al.  On the toric algebra of graphical models , 2006, math/0608054.