By Sumio Watanabe
Bound to be influential, Watanabe's booklet lays the principles for using algebraic geometry in statistical studying concept. Many models/machines are singular: combination types, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the idea completed the following underpins exact estimation ideas within the presence of singularities.
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Additional info for Algebraic Geometry and Statistical Learning Theory
Z(ξ ) Therefore four errors are given by the linear sums of two expectations of Sλ (ξ (u)) and Sλ (ξ (u)). By eliminating two expectations from four equations, we obtain two equations which hold for the Bayes quartet. 38 Introduction Main Formula III (Equations of states in statistical estimation) There are two universal relations in Bayes quartet. 28) E[G∗g ] − E[G∗t ] = 2β(E[G∗t ] − E[Bt∗ ]). 29) These equations hold for an arbitrary true distribution, an arbitrary statistical model, an arbitrary a priori distribution, and arbitrary singularities.
In this chapter, the definition of singularities and the basic theorem for resolution of singularities are introduced. 6. 1 Polynomials and analytic functions Let d be a natural number. Let R and C be the set of all real numbers and the set of all complex numbers respectively. A d-dimensional multi-index α is defined by α = (α1 , α2 , . . , αd ), where α1 , α2 , . . , αd are nonnegative integers. For given x, b ∈ Rd x = (x1 , x2 , . . , xd ), b = (b1 , b2 , . . , bd ), and aα ∈ R, we define aα (x − b)α = aα1 α2 ···αd (x1 − b1 )α1 (x2 − b2 )α2 · · · (xd − bd )αd .
Moreover, Main Formula III holds even if the true distribution is not contained in the model . 4 ML and MAP theory The last formula concerns the maximum likelihood or a posteriori method. Let W be a compact set, and f (x, w) and ϕ(w) be respectively analytic and C 2 -class functions of w ∈ W . Then there exists a parameter wˆ ∈ W that minimizes the generalized log likelihood ratio function, n Rn0 (w) = f (Xi , w) − an log ϕ(w), i=1 where an is a nondecreasing sequence. Note that, if W is not compact, the parameter that minimizes Rn0 (w) does not exist in general.
Algebraic Geometry and Statistical Learning Theory by Sumio Watanabe