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Watanabe S. Algebraic Geometry and Statistical Learning Theory

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Watanabe S. Algebraic Geometry and Statistical Learning Theory
Cambridge: Cambridge University Press, 2009. — 296 p.
Contents
Preface
Random samples
Kullback–Leibler distance
Fisher information matrix
Singular models
Density estimation
Conditional probability density
Statistical estimation methods
Evidence
Bayes and Gibbs estimations
Maximum likelihood and a posteriori
Standard form of log likelihood ratio function
Evidence of singular model
Bayes and Gibbs theory
ML and MAP theory
Overview of this book
Probability theory
Polynomials and analytic functions
Algebraic set and analytic set
Singularity
Resolution of singularities
Normal crossing singularities
Manifold
Ring and ideal
Real algebraic set
Singularities and dimension
Real projective space
Blow-up
Simple cases
A sample of a statistical model
Schwartz distribution
State density function
Mellin transform
Evaluation of singular integral
Asymptotic expansion and b-function
Convergence in law
Function-valued analytic functions
Empirical process
Fluctuation of Gaussian processes
Singular learning theory
Standard form of likelihood ratio function
Evidence and stochastic complexity
Equations of states
Basic lemmas
Proof of the theorems
Maximum likelihood and a posteriori
Learning coefficient
Three-layered neural networks
Mixture models
Bayesian network
Hidden Markov model
Singular learning process
Bias and variance
Non-analytic learning machines
Universally optimal learning
Generalized Bayes information criterion
Widely applicable information criteria
Experiments
Optimal hypothesis test
Example of singular hypothesis test
Markov chain Monte Carlo
Metropolis Algorithm
Variational Bayes approximation
From regular to singular
Bibliography
Index
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