An introduction to algebraic statistics with tensors

The Treatment of Random Variables......Page 7
  Bernoulli Binary Models......Page 9
  Mixture Models......Page 11
  Conclusion......Page 13
Contents......Page 14
About the Authors......Page 17
Part I Algebraic Statistics......Page 18
 1.1 Systems of Random Variables......Page 19
 1.2 Distributions......Page 23
 1.3 Measurements on a Distribution......Page 27
 1.4 Exercises......Page 29
 2.1 Basic Probability......Page 30
 2.2 Booleanization and Logic Connectors......Page 34
 2.3 Independence Connections and Marginalization......Page 36
 2.4 Exercises......Page 49
 3.1 Models......Page 50
 3.2 Independence Models......Page 51
 3.3 Connections and Parametric Models......Page 53
 3.4 Toric Models and Exponential Matrices......Page 58
 4.1 Motivations......Page 61
 4.2 Projective Algebraic Models......Page 63
 4.3 Parametric Models......Page 64
5 Conditional Independence......Page 68
 5.1 Models of Conditional Independence......Page 71
 5.2 Markov Chains and Trees......Page 78
 5.3 Hidden Variables......Page 83
 References......Page 90
Part II Multi-linear Algebra......Page 91
 6.1 Basic Definitions......Page 92
 6.2 The Tensor Product......Page 98
 6.3 Rank of Tensors......Page 104
 6.4 Tensors of Rank 1......Page 106
 6.5 Exercises......Page 113
 7.1 Generalities and Examples......Page 115
 7.2 The Rank of a Symmetric Tensor......Page 116
 7.3 Symmetric Tensors and Polynomials......Page 120
 7.4 The Complexity of Polynomials......Page 123
 References......Page 125
 8.1 Marginalization......Page 127
 8.2 Contractions......Page 131
 8.3 Scan and Flattening......Page 134
 Reference......Page 139
Part III Commutative Algebra and Algebraic Geometry......Page 140
 9.1 Projective Varieties......Page 141
  9.1.1 Associated Ideals......Page 146
  9.1.2 Topological Properties of Projective Varieties......Page 149
 9.2 Multiprojective Varieties......Page 153
 9.3 Projective and Multiprojective Maps......Page 156
 9.4 Exercises......Page 161
 References......Page 162
 10.1 Linear Maps and Change of Coordinates......Page 163
 10.2 Elimination Theory......Page 166
 10.3 Forgetting a Variable......Page 169
 10.4 Linear Projective and Multiprojective Maps......Page 171
 10.5 The Veronese Map and the Segre Map......Page 172
 10.6 The Chow's Theorem......Page 182
 10.7 Exercises......Page 184
 Reference......Page 185
 11.1 Complements on Irreducible Varieties......Page 186
 11.2 Dimension......Page 187
 11.3 General Maps......Page 196
 11.4 Exercises......Page 199
 References......Page 200
 12.1 Definitions......Page 201
  12.2.1 Tangent Spaces and the Terracini's Lemma......Page 209
  12.2.2 Inverse Systems......Page 212
 12.3 Exercises......Page 215
 13.1 Monomial Orderings......Page 216
 13.2 Monomial Ideals......Page 222
 13.3 Groebner Basis......Page 226
 13.4 Buchberger's Algorithm......Page 231
 13.5 Groebner Bases and Elimination Theory......Page 233
 References......Page 237
Index......Page 238