Lectures on Algebraic Statistics

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Springer, 2009 - Mathematics - 171 pages
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How does an algebraic geometer studying secant varieties further the understanding of hypothesis tests in statistics? Why would a statistician working on factor analysis raise open problems about determinantal varieties? Connections of this type are at the heart of the new field of "algebraic statistics". In this field, mathematicians and statisticians come together to solve statistical inference problems using concepts from algebraic geometry as well as related computational and combinatorial techniques. The goal of these lectures is to introduce newcomers from the different camps to algebraic statistics. The introduction will be centered around the following three observations: many important statistical models correspond to algebraic or semi-algebraic sets of parameters; the geometry of these parameter spaces determines the behaviour of widely used statistical inference procedures; computational algebraic geometry can be used to study parameter spaces and other features of statistical models.

  

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Contents

Markov Bases
1
12 Markov Bases of Hierarchical Models
11
13 The Many Bases of an Integer Lattice
19
Likelihood Inference
29
22 Likelihood Equations for Implicit Models
40
23 Likelihood Ratio Tests
48
Conditional Independence
60
32 Graphical Models
69
52 Exact Integration for Discrete Models
114
Exercises
123
62 Quasisymmetry and Cycles
128
63 A Colored Gaussian Graphical Model
131
64 Instrumental Variables and Tangent Cones
135
65 Fisher Information for Multivariate Normals
142
66 The Intersection Axiom and Its Failure
144
67 Primary Decomposition for CI Inference
147

33 Parametrizations of Graphical Models
79
Hidden Variables
89
42 Factor Analysis
99
Bayesian Integrals
105
68 An Independence Model and Its Mixture
150
Open Problems
157
Bibliography
164
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