 | Aoki S. Markov bases in algebraic statistics / S.Aoki, H.Hara, A.Takemura. - New York: Springer, 2012. - xi, 298 p.: ill. - (Springer series in statistics). - Ref.: p. 287-293. - Ind.: p.295-298. - ISBN 978-1-4614-3718-5
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Part I Introduction and Some Relevant Preliminary Material
1 Exact Tests for Contingency Tables and Discrete
Exponential Families ......................................... 3
1.1 Independence Model of 2 × 2 Two-Way Contingency
Tables .................................................. 3
1.2 2x2 Contingency Table Models as Discrete Exponential
Family .................................................. 8
1.3 Independence Model of General Two-Way Contingency
Tables ................................................. 10
1.4 Conditional Independence Model of Three-Way
Contingency Tables ..................................... 14
1.4.1 Normalizing Constant of Hypergeometric
Distribution for the Conditional Independence
Model ........................................... 18
1.5 Notation of Hierarchical Models for m-Way
Contingency Tables ..................................... 19
2 Markov Chain Monte Carlo Methods over Discrete
Sample Space ................................................ 23
2.1 Constructing a Connected Markov Chain over
a Conditional Sample Space: Markov Basis ............... 23
2.2 Adjusting Transition Probabilities by Metropolis-
Hastings Algorithm ..................................... 27
3 Toric Ideals and Their Gröbner Bases ........................ 33
3.1 Polynomial Ring ........................................ 33
3.2 Term Order and Grobner Basis ........................... 35
3.3 Buchberger's Algorithm ................................. 38
3.4 Elimination Theory ..................................... 39
3.5 Toric Ideals ........................................... 39
Part II Properties of Markov Bases
4 Definition of Markov Bases and Other Bases .................. 47
4.1 Discrete Exponential Family ............................ 47
4.2 Definition of Markov Basis ............................. 50
4.3 Properties of Moves and the Lattice Basis .............. 51
4.4 The Fundamental Theorem of Markov Basis ................ 54
4.5 Grobner Basis from the Viewpoint of Markov Basis ....... 59
4.6 Graver Basis, Lawrence Lifting, and Logistic
Regression ............................................. 60
5 Structure of Minimal Markov Bases ........................... 65
5.1 Accessibility by a Set of Moves ........................ 65
5.2 Structure of Minimal Markov Basis and Indispensable
Moves .................................................. 66
5.3 Minimum Fiber Markov Basis ............................. 71
5.4 Examples of Minimal Markov Bases ....................... 72
5.4.1 One-Way Contingency Tables ...................... 72
5.4.2 Independence Model of Two-Way Contingency
Tables .......................................... 73
5.4.3 The Unique Minimal Markov Basis for the
Lawrence Lifting ................................ 73
5.5 Indispensable Monomials ................................ 75
6 Method of Distance Reduction ................................ 79
6.1 Distance Reducing Markov Bases ......................... 79
6.2 Examples of Distance-Reducing Proofs ................... 81
6.2.1 The Complete Independence Model of Three-Way
Contingency Tables .............................. 81
6.2.2 Hardy-Weinberg Model ............................ 83
6.3 Graver Basis and 1-Norm Reducing Markov Bases .......... 85
6.4 Some Results on Minimality of 1-Norm Reducing
Markov Bases ........................................... 86
7 Symmetry of Markov Bases .................................... 91
7.1 Motivations for Invariance of Markov Bases ............. 91
7.2 Examples of Invariant Markov Bases ..................... 92
7.3 Action of Symmetric Group on the Set of Cells .......... 93
7.4 Symmetry of a Toric Model and the Largest Group
of Invariance .......................................... 96
7.5 The Largest Group of Invariance for the Independence
Model of Two-Way Tables ................................ 98
7.6 Characterizations of a Minimal Invariant Markov
Basis ................................................. 100
Part III Markov Bases for Specific Models
8 Decomposable Models of Contingency Tables .................. 109
8.1 Chordal Graphs and Decomposable Models ................ 109
8.2 Markov Bases for Decomposable Models .................. 111
8.3 Structure of Degree 2 Fibers .......................... 113
8.4 Minimal Markov Bases for Decomposable Models .......... 115
8.5 Minimal Invariant Markov Bases ........................ 119
8.6 The Relation Between Minimal and Minimal Invariant
Markov Bases .......................................... 127
9 Markov Basis for No-Three-Factor Interaction Models
and Some Other Hierarchical Models ......................... 129
9.1 No-Three-Factor Interaction Models for 3 × 3 × К
Contingency Tables .................................... 129
9.2 Unique Minimal Markov Basis for 3 × 3 × 3 Tables ...... 130
9.3 Unique Minimal Markov Basis for 3 × 3 × 4 Tables ...... 139
9.4 Unique Minimal Markov Basis for 3 × 3 × 5 and
3 × 3 × К Tables for K > 5 ............................ 142
9.5 Indispensable Moves for Larger Tables ................. 145
9.6 Reducible Models ...................................... 149
9.7 Markov Basis for Reducible Models ..................... 150
9.8 Markov Complexity and Graver Complexity ............... 153
9.9 Markov Width for Some Hierarchical Models ............. 156
10 Two-Way Tables with Structural Zeros and Fixed Subtable
Sums ....................................................... 159
10.1 Markov Bases for Two-Way Tables with Structural
Zeros ................................................. 159
10.1.1 Quasi-Independence Model in Two-Way
Incomplete Contingency Tables .................. 159
10.1.2 Unique Minimal Markov Basis for Two-Way
Quasi-Independence Model ....................... 161
10.1.3 Enumerating Elements of the Minimal Markov
Basis .......................................... 164
10.1.4 Numerical Example of a Quasi-Independence
Model .......................................... 167
10.2 Markov Bases for Subtable Sum Problem ................. 168
10.2.1 Introduction of Subtable Sum Problem ........... 168
10.2.2 Markov Bases Consisting of Basic Moves ......... 169
10.2.3 Markov Bases for Common Diagonal Effect
Models ......................................... 172
10.2.4 Numerical Examples of Common Diagonal
Effect Models .................................. 176
11 Regular Factorial Designs with Discrete Response
Variables .................................................. 181
11.1 Conditional Tests for Designed Experiments with
Discrete Observations ................................. 181
11.1.1 Conditional Tests for Log-Linear Models of
Poisson Observations ........................... 181
11.1.2 Models and Aliasing Relations .................. 184
11.1.3 Conditional Tests for Logistic Models of
Binomial Observations .......................... 191
11.1.4 Example: Wave-Soldering Data ................... 193
11.2 Markov Bases and Corresponding Models for
Contingency Tables .................................... 194
11.2.1 Rewriting Observations as Frequencies of a
Contingency Table .............................. 194
11.2.2 Models for the Two-Level Regular Fractional
Factorial Designs with 16 Runs ................. 200
11.2.3 Three-Level Regular Fractional Factorial
Designs and 3s-k Continent Tables .............. 203
12 Groupwise Selection Models ................................. 209
12.1 Examples of Groupwise Selections ...................... 209
12.1.1 The Case of National Center Test in Japan ...... 209
12.1.2 The Case of Hardy-Weinberg Models for Allele
Frequency Data ................................. 212
12.2 Conditional Tests for Groupwise Selection Models ...... 213
12.2.1 Models for NCT Data ............................ 214
12.2.2 Models for Allele Frequency Data ............... 215
12.3 Grobner Basis for Segre-Veronese Configuration ........ 217
12.4 Sampling from the Grobner Basis for the Segre-
Veronese Configuration ................................ 219
12.5 Numerical Examples .................................... 219
12.5.1 The Analysis of NCT Data ....................... 219
12.5.2 The Analysis of Allele Frequency Data .......... 221
13 The Set of Moves Connecting Specific Fibers ................ 229
13.1 Discrete Logistic Regression Model with One
Covariate ............................................. 229
13.2 Discrete Logistic Regression Model with More than
One Covariate ......................................... 231
13.3 Numerical Examples .................................... 238
13.3.1 Exact Tests of Logistic Regression Model ....... 238
13.4 Connecting Zero-One Tables with Graver Basis .......... 240
13.5 Rasch Model ........................................... 241
13.6 Many-Facet Rasch Model ................................ 242
13.7 Latin Squares and Zero-One Tables for No-Three-
Factor Interaction Models ............................. 245
Part IV Some Other Topics of Algebraic Statistics
14 Disclosure Limitation Problem and Markov Basis ............. 251
14.1 Swapping with Some Marginals Fixed .................... 251
14.2 E-Swapping ............................................ 252
14.3 Equivalence of Degree-Two Square-Free Move of Markov
Bases and Swapping of Two Records ..................... 253
14.4 Swappability Between Two Records ...................... 254
14.5 Searching for Another Record for Swapping ............. 257
15 Grobner Basis Techniques for Design of Experiments ......... 261
15.1 Design Ideals ......................................... 261
15.2 Identifiability of Polynomial Models and the
Quotient with Respect to the Design Ideal ............. 262
15.3 Regular Two-Level Designs ............................. 267
15.4 Indicator Functions ................................... 269
16 Running Markov Chain Without Markov Bases .................. 275
16.1 Performing Conditional Tests When a Markov Basis
Is Not Available ...................................... 275
16.2 Sampling Contingency Tables with a Lattice Basis ...... 275
16.3 A Lattice Basis for Higher Lawrence Configuration ..... 277
16.4 Numerical Experiments ................................. 278
16.4.1 No-Three-Factor Interaction Model .............. 278
16.4.2 Discrete Logistic Regression Model ............. 282
References .................................................... 287
Index ......................................................... 295
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