Algebraic and geometric methods in statistics

This up-to-date account of algebraic statistics and information geometry explores the emerging connections between the two disciplines, demonstrating how they can be used in design of experiments and how they benefit our understanding of statistical models, in particular, exponential models. This book presents a new way of approaching classical statistical problems and raises scientific questions that would never have been considered without the interaction of these two disciplines. Beginning with a brief introduction to each area, using simple illustrative examples, the book then proceeds with a collection of reviews and some new results written by leading researchers in their respective fields. Part III dwells in both classical and quantum information geometry, containing surveys of key results and new material. Finally, Part IV provides examples of the interplay between algebraic statistics and information geometry. Computer code and proofs are also available online, where key examples are developed in further detail.

[1]  S. Luo.  Wigner-Yanase skew information vs. quantum Fisher information , 2003 .

[2]  Paolo Gibilisco,et al.  A CHARACTERISATION OF WIGNER YANASE SKEW INFORMATION AMONG STATISTICALLY MONOTONE METRICS , 2001 .

[3]  E. Wigner.  Die Messung quantenmechanischer Operatoren , 1952 .

[4]  Eugen Fick,et al.  The quantum statistics of dynamic processes , 1990 .

[5]  M. Casanellas,et al.  Performance of a new invariants method on homogeneous and nonhomogeneous quartet trees. , 2006, Molecular biology and evolution.

[6]  Giovanni Pistone,et al.  Finitely Generated Cumulants , 1999 .

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

[8]  Michael Francis Atiyah,et al.  Introduction to commutative algebra , 1969 .

[9]  S. Luo,et al.  Quantum versus classical uncertainty , 2005 .

[10]  B. Efron.  Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency) , 1975 .

[11]  Seth Sullivant,et al.  Combinatorial secant varieties , 2005 .

[12]  Terence P. Speed,et al.  Invariants of Some Probability Models Used in Phylogenetic Inference , 1993 .

[13]  Jan Naudts,et al.  Linear response theory and the KMS condition , 1975 .

[14]  T. Isola,et al.  Uncertainty principle and quantum Fisher information. II. , 2007 .

[15]  Attila Andai,et al.  Uncertainty principle with quantum Fisher information , 2007, 0707.1147.

[16]  M. Drton,et al.  Algebraic factor analysis: tetrads, pentads and beyond , 2005, math/0509390.

[17]  Frank Hansen,et al.  Inequalities for Quantum Skew Information , 2008 .

[18]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[19]  Henry P. Wynn,et al.  Generalised confounding with Grobner bases , 1996 .

[20]  Frank Hansen.  Metric adjusted skew information , 2008, Proceedings of the National Academy of Sciences.

[21]  D. Petz,et al.  Geometries of quantum states , 1996 .

[22]  S. Eguchi.  Second Order Efficiency of Minimum Contrast Estimators in a Curved Exponential Family , 1983 .

[23]  Giovanni Pistone,et al.  Algebraic Bayesian analysis of contingency tables with possibly zero-probability cells , 2006 .

[24]  Frank Hansen.  The Wigner-Yanase Entropy is not Subadditive , 2007 .

[25]  P. Diaconis,et al.  Algebraic algorithms for sampling from conditional distributions , 1998 .

[26]  T. Isola,et al.  On a correspondence between regular and non-regular operator monotone functions , 2008, 0808.0468.

[27]  A. Dawid,et al.  Game theory, maximum entropy, minimum discrepancy and robust Bayesian decision theory , 2004, math/0410076.

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

[29]  S. Luo.  Wigner-Yanase skew information and uncertainty relations. , 2003, Physical review letters.

[30]  H. Araki,et al.  Measurement of Quantum Mechanical Operators , 1960 .

[31]  Frank Hansen.  Characterizations of symmetric monotone metrics on the state space of quantum systems , 2006, Quantum Inf. Comput..

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

[33]  E. Wigner,et al.  INFORMATION CONTENTS OF DISTRIBUTIONS. , 1963, Proceedings of the National Academy of Sciences of the United States of America.

[34]  Satoshi Aoki,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS The largest group of invariance for Markov bases and toric ideals , 2005 .

[35]  D. Petz.  Monotone metrics on matrix spaces , 1996 .