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.
General
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