02161nam a22003138i 4500001001600000003000700016008004100023020001800064020001800082035002000100041000800120082001600128100002900144245008400173264007100257300004000328336002600368337002600394338003600420490007200456500001300528520102700541650005601568650002501624856005501649932003201704596000601736949010501742CR9780511800474UkCbUP101021s2009||||enk o ||1 0|eng|d a9780511800474 a9780521864671 a(Sirsi) a792373 aeng00a006.3/12221 aWatanabe, Sumio,eauthor10aAlgebraic geometry and statistical learning theoryh[E-Book] /cSumio Watanabe. 1aCambridge :bCambridge University Press,c2009e(CUP)fCUP20200108 a1 online resource (viii, 286 pages) atextbtxt2rdacontent acomputerbc2rdamedia aonline resourcebcr2rdacarrier aCambridge monographs on applied and computational mathematics ;v25 aenglisch aSure 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. 0aComputational learning theoryxStatistical methods. 0aGeometry, Algebraic.40uhttps://doi.org/10.1017/CBO9780511800474zVolltext aCambridgeCore (Order 30059) a1 aXX(792373.1)wAUTOc1i792373-1001lELECTRONICmZBrNsYtE-BOOKu8/1/2020xUNKNOWNzUNKNOWN1ONLINE