Algebraic geometry and statistical learning theory
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 n...
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| Format: | E-Book |
| Sprache: | English |
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Cambridge University Press
2009
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| Schriftenreihe: | Cambridge monographs on applied and computational mathematics
25 |
| Online-Zugang: | Volltext |
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| 100 | 1 | |a Watanabe, Sumio |d 1959- | |
| 245 | 1 | 0 | |a Algebraic geometry and statistical learning theory |c Sumio Watanabe |
| 246 | 3 | |a Algebraic Geometry & Statistical Learning Theory | |
| 264 | 1 | |a Cambridge |b Cambridge University Press |c 2009 | |
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| 490 | 1 | |a Cambridge monographs on applied and computational mathematics |v 25 | |
| 520 | |a 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. | ||
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| id | ZDB-20-CTM-CR9780511800474 |
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| indexdate | 2024-12-18T12:04:30Z |
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| series2 | Cambridge monographs on applied and computational mathematics |
| spelling | Watanabe, Sumio 1959- Algebraic geometry and statistical learning theory Sumio Watanabe Algebraic Geometry & Statistical Learning Theory Cambridge Cambridge University Press 2009 1 Online-Ressource (viii, 286 Seiten) txt c cr Cambridge monographs on applied and computational mathematics 25 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. Erscheint auch als Druck-Ausgabe 9780521864671 TUM01 ZDB-20-CTM TUM_PDA_CTM https://doi.org/10.1017/CBO9780511800474 Volltext |
| spellingShingle | Watanabe, Sumio 1959- Algebraic geometry and statistical learning theory |
| title | Algebraic geometry and statistical learning theory |
| title_alt | Algebraic Geometry & Statistical Learning Theory |
| title_auth | Algebraic geometry and statistical learning theory |
| title_exact_search | Algebraic geometry and statistical learning theory |
| title_full | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_fullStr | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_full_unstemmed | Algebraic geometry and statistical learning theory Sumio Watanabe |
| title_short | Algebraic geometry and statistical learning theory |
| title_sort | algebraic geometry and statistical learning theory |
| url | https://doi.org/10.1017/CBO9780511800474 |
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