Format:
1 Online-Ressource (viii, 286 pages)
,
digital, PDF file(s)
ISBN:
9780511800474
Series Statement:
Cambridge monographs on applied and computational mathematics 25
Content:
Sure to be influential, this book lays the foundations for the use of algebraic geometry in statistical learning theory. Many widely used statistical models and learning machines applied to information science have a parameter space that is singular: mixture models, neural networks, HMMs, Bayesian networks, and stochastic context-free grammars are major examples. Algebraic geometry and singularity theory provide the necessary tools for studying such non-smooth models. Four main formulas are established: 1. the log likelihood function can be given a common standard form using resolution of singularities, even applied to more complex models; 2. the asymptotic behaviour of the marginal likelihood or 'the evidence' is derived based on zeta function theory; 3. new methods are derived to estimate the generalization errors in Bayes and Gibbs estimations from training errors; 4. the generalization errors of maximum likelihood and a posteriori methods are clarified by empirical process theory on algebraic varieties
Note:
Title from publisher's bibliographic system (viewed on 05 Oct 2015)
Additional Edition:
ISBN 9780521864671
Additional Edition:
Print version ISBN 9780521864671
Additional Edition:
Erscheint auch als Druck-Ausgabe Watanabe, Sumio, 1959 - Algebraic geometry and statistical learning theory Cambridge : Cambridge University Press, 2009 ISBN 9780521864671
Additional Edition:
ISBN 0521864674
Language:
English
Subjects:
Economics
,
Mathematics
Keywords:
Algebraische Geometrie
;
Mathematische Lerntheorie
DOI:
10.1017/CBO9780511800474
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