Algebraic Geometry and Real-World Information Science

摘要

Nearly all the probabilistic models in information science, statistical inference, and gene analysis are singular; that is, their Fisher information matrix is singular. Hierarchical learning machines, normal mixtures, Bayesian networks, hidden Markov models, and probabilistic context-free grammars are singular. The conventional statistical theorems never hold in singular models. The maximum-likelihood estimator fails to satisfy asymptotic normality. Nor does the Bayes a posteriori distribution converge in a normal distribution.