Written by pioneers in this exciting new field, "Algebraic Statistics" introduces the application of polynomial algebra to experimental design, discrete probability, and statistics. (Mathematics)

Giovanni Pistone is a Professor of Mathematical Statistics in the Mathematics Department at the Politecnico, Torino, Italy.

List of figures

ix

List of tables

xi

Preface

xiii

Notation

xv

Introduction

1

(14)

Outline

1

(4)

Computer Algebra

5

(5)

An example: the 2 3-1 fractional factorial design

10

(5)

Algebraic models

15

(28)

Models

16

(1)

Polynomial ideals

17

(2)

Term-orderings

19

(3)

Division algorithm

22

(1)

Hilbert basis theorem

23

(2)

Varieties and equations

25

(2)

Grobner bases

27

(2)

Properties of a Grobner basis

29

(2)

Elimination theory

31

(2)

Polynomial functions and quotients by ideals

33

(2)

Hilbert function

35

(1)

Further topics

36

(7)

Grobner bases in experimental design

43

(32)

Designs and design ideals

43

(1)

Computing the Grobner basis of a design

44

(3)

Operations with designs

47

(1)

Examples

48

(2)

Span of a design

50

(3)

Models and identifiability: quotients

53

(1)

Confounding of models

54

(2)

Further examples

56

(4)

The fan of an experimental design

60

(3)

Minimal and maximal fan designs

63

(2)

Hilbert functions and fans for graded ordering

65

(1)

Subsets and algorithms

66

(5)

Regression analysis

71

(1)

Non-polynomial models

72

(3)

Two-level factors: logic, reliability, design

75

(20)

The binary case: Boolean representations

75

(3)

Grobner bases and Boolean ideals

78

(2)

Logic and learning

80

(1)

Reliability: coherent systems as minimal fan designs

81

(2)

Inclusion-exclusion and tube theory

83

(7)

Two-level factorial design: contrasts and orthogonality

90

(5)

Probability

95

(24)

Random variables on a finite support

96

(1)

The ring of random variables

97

(2)

Matrix representation of ≈(D,K)

99

(2)

Uniform probability

101

(2)

Probability densities

103

(3)

Image probability and marginalisation

106

(2)

Conditional expectation

108

(3)

Algebraic representation of exponentials

111

(2)

Exponential form of a probability

113

(6)

Statistical modeling

119

(36)

Introduction

119

(1)

Statistical models

120

(8)

Generating functions and exponential submodels

128

(3)

Likelihoods and sufficient statistics

131

(4)

Score function and information

135

(1)

Estimation: lattice case

136

(2)

Finitely generated cumulants

138

(1)

Estimating functions

139

(1)

An extended example

140

(9)

Orthogonality and toric ideals

149

(6)

References

155

(4)

Index

159

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