Lectures on Algebraic Statistics

Markov Bases.- Likelihood Inference.- Conditional Independence.- Hidden Variables.- Bayesian Integrals.- Exercises.- Open Problems.

[1]  Milan Studený,et al.  A Graphical Representation of Equivalence Classes of AMP Chain Graphs , 2006, J. Mach. Learn. Res..

[2]  D. Madigan,et al.  A characterization of Markov equivalence classes for acyclic digraphs , 1997 .

[3]  R. Stanley.  What Is Enumerative Combinatorics , 1986 .

[4]  M. Drton.  Likelihood ratio tests and singularities , 2007, math/0703360.

[5]  M. Frydenberg.  The chain graph Markov property , 1990 .

[6]  Seth Sullivant,et al.  Algebraic geometry of Gaussian Bayesian networks , 2007, Adv. Appl. Math..

[7]  Benjamin Georgi,et al.  Context-specific independence mixture modeling for positional weight matrices , 2006, ISMB.

[8]  J. Landsberg,et al.  On the ideals and singularities of secant varieties of Segre varieties , 2006, math/0601452.

[9]  Milan Studený,et al.  Probabilistic conditional independence structures , 2006, Information science and statistics.

[10]  L. Pachter,et al.  Tropical geometry of statistical models. , 2003, Proceedings of the National Academy of Sciences of the United States of America.

[11]  Michael Evans,et al.  Latent class analysis of two-way contingency tables by Bayesian methods , 1989 .

[12]  J. Darroch,et al.  Generalized Iterative Scaling for Log-Linear Models , 1972 .

[13]  N. Wermuth,et al.  Joint response graphs and separation induced by triangular systems , 2004 .

[14]  A. Takemura,et al.  Minimal Basis for a Connected Markov Chain over 3 × 3 ×K Contingency Tables with Fixed Two‐Dimensional Marginals , 2003 .

[15]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[16]  David Mond,et al.  Stochastic factorizations, sandwiched simplices and the topology of the space of explanations , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[17]  D. Haughton.  On the Choice of a Model to Fit Data from an Exponential Family , 1988 .

[18]  Fabio Rapallo.  Algebraic Markov Bases and MCMC for Two‐Way Contingency Tables , 2003 .

[19]  Bernd Sturmfels,et al.  Open Problems in Algebraic Statistics , 2007, 0707.4558.

[20]  Seth Sullivant,et al.  Toric Ideals of Phylogenetic Invariants , 2004, J. Comput. Biol..

[21]  E. Allman,et al.  Phylogenetic invariants for the general Markov model of sequence mutation. , 2003, Mathematical biosciences.

[22]  Sumio Watanabe,et al.  Algebraic geometry and stochastic complexity of hidden Markov models , 2005, Neurocomputing.

[23]  Bernd Sturmfels,et al.  Higher Lawrence configurations , 2003, J. Comb. Theory, Ser. A.

[24]  Seth Sullivant,et al.  Algebraic statistical models , 2007 .

[25]  Jan Draisma,et al.  On the ideals of equivariant tree models , 2007, 0712.3230.

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

[27]  Seth Sullivant,et al.  Gröbner Bases and Polyhedral Geometry of Reducible and Cyclic Models , 2002, J. Comb. Theory, Ser. A.

[28]  Herman Rubin,et al.  Statistical Inference in Factor Analysis , 1956 .

[29]  Bernd Sturmfels,et al.  Solving the Likelihood Equations , 2005, Found. Comput. Math..

[30]  渡邊 澄夫.  Algebraic geometry and statistical learning theory , 2009 .

[31]  Bernd Sturmfels,et al.  Algebraic geometry of Bayesian networks , 2005, J. Symb. Comput..

[32]  D. Edwards.  Introduction to graphical modelling , 1995 .

[33]  Roderick Wong,et al.  Asymptotic approximations of integrals , 1989, Classics in applied mathematics.

[34]  Stephen A. Vavasis,et al.  On the Complexity of Nonnegative Matrix Factorization , 2007, SIAM J. Optim..

[35]  O. Barndorff-Nielsen.  Information And Exponential Families , 1970 .

[36]  Sumio Watanabe,et al.  Algebraic Analysis for Nonidentifiable Learning Machines , 2001, Neural Computation.

[37]  Erwin Kreyszig,et al.  Introductory Mathematical Statistics. , 1970 .

[38]  Linda June Davis.  Exact tests for 2×2 contingency tables , 1986 .

[39]  Prolongations and Computational Algebra , 2006, Canadian Journal of Mathematics.

[40]  Akimichi Takemura,et al.  MATHEMATICAL ENGINEERING TECHNICAL REPORTS Markov Bases for Two-way Subtable Sum Problems , 2007, 0708.2312.

[41]  P. Spirtes,et al.  Ancestral graph Markov models , 2002 .

[42]  N. Wermuth,et al.  Graphical Models for Associations between Variables, some of which are Qualitative and some Quantitative , 1989 .

[43]  M. Perlman,et al.  Characterizing Markov equivalence classes for AMP chain graph models , 2006, math/0607037.

[44]  Christian P. Robert,et al.  Monte Carlo Statistical Methods (Springer Texts in Statistics) , 2005 .

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

[46]  Aldo Conca,et al.  Gröbner Bases of Ideals of Minors of a Symmetric Matrix , 1994 .

[47]  D. Madigan,et al.  Alternative Markov Properties for Chain Graphs , 2001 .

[48]  P. Diaconis,et al.  Testing for independence in a two-way table , 1985 .

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

[50]  Seth Sullivant.  Toric fiber products , 2006 .

[51]  Seth Sullivant,et al.  A finiteness theorem for Markov bases of hierarchical models , 2007, J. Comb. Theory, Ser. A.

[52]  H. Chernoff.  On the Distribution of the Likelihood Ratio , 1954 .

[53]  Seth Sullivant,et al.  Markov models for accumulating mutations , 2007, 0709.2646.

[54]  D. Eisenbud.  Commutative Algebra: with a View Toward Algebraic Geometry , 1995 .

[55]  On the Gr\"obner complexity of matrices , 2007, 0708.4392.

[56]  Bernd Sturmfels,et al.  Marginal Likelihood Integrals for Mixtures of Independence Models , 2008, J. Mach. Learn. Res..

[57]  Sumio Watanabe,et al.  Stochastic complexities of reduced rank regression in Bayesian estimation , 2005, Neural Networks.

[58]  S. Sullivant,et al.  Markov Bases of Binary Graph Models , 2003, math/0308280.

[59]  Jesús A. De Loera,et al.  Markov bases of three-way tables are arbitrarily complicated , 2006, J. Symb. Comput..

[60]  Sumio Watanabe,et al.  Singularities in mixture models and upper bounds of stochastic complexity , 2003, Neural Networks.

[61]  G. Schwarz.  Estimating the Dimension of a Model , 1978 .

[62]  Søren Højsgaard,et al.  Graphical Gaussian models with edge and vertex symmetries , 2008 .

[63]  Akimichi Takemura,et al.  A Markov basis for conditional test of common diagonal effect in quasi-independence model for square contingency tables , 2009, Comput. Stat. Data Anal..

[64]  Ronald Christensen,et al.  Log-Linear Models and Logistic Regression , 1997 .

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

[66]  F. Zak.  Tangents and Secants of Algebraic Varieties , 1993 .

[67]  Rekha R. Thomas,et al.  Gröbner bases and triangulations of the second hypersimplex , 1995, Comb..

[68]  J. M. Landsberg,et al.  On the Ideals of Secant Varieties of Segre Varieties , 2004, Found. Comput. Math..

[69]  T. L. Kelley.  Essential Traits of Mental Life , 2012, Nature.

[70]  Lawrence D. Brown.  Fundamentals of Statistical Exponential Families , 1987 .

[71]  Stephen E. Fienberg,et al.  Discrete Multivariate Analysis: Theory and Practice , 1976 .

[72]  Dan Geiger,et al.  Asymptotic Model Selection for Naive Bayesian Networks , 2002, J. Mach. Learn. Res..

[73]  A. Dobra.  Markov bases for decomposable graphical models , 2003 .

[74]  G. Ziegler.  Lectures on Polytopes , 1994 .

[75]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[76]  Wolfgang Vogel,et al.  Joins and intersections , 1991 .

[77]  Ingram Olkin,et al.  Moments of minors of Wishart matrices , 2006 .