By Sumio Watanabe
Certain to be influential, Watanabe's booklet lays the rules for using algebraic geometry in statistical studying thought. Many models/machines are singular: blend types, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the idea completed right here underpins actual estimation innovations within the presence of singularities.
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Additional info for Algebraic Geometry and Statistical Learning Theory
Xn are independently subject to the same distribution as X. As the central limit theorem is characterized by the mean and the variance of the random variables, the statistical learning theory is characterized by the largest pole of the zeta function and the singular fluctuation. The large deviation theory indicates that Fn /n → pS, where S is the entropy of X. Main Formulas II and III show more precise results than the large deviation theory. 2 Singularity theory A lot of statistical models and learning machines contain singularities in their parameter spaces.
By using the convergence in law ξn (u) → ξ (u), the largest term of the asymptotic expansion of the a posteriori distribution is given by (log n)m−1 Yn (w) dw ∼ du∗ = nλ ∞ √ dt t λ−1 e−βt+ tβ ξ (u) . 25) 0 The normalized evidence is Zn0 = Yn (w) dw. 26) 34 Introduction where F R (ξ ) is a random variable F R (ξ ) = − log ∞ du∗ √ dt t λ−1 e−βt+ tβ ξ (u) . 0 We obtain the second main result. Main Formula II (Convergence of stochastic complexity) Let (−λ) and m be respectively the largest pole and its order of the zeta function ζ (z) = K(w)z ϕ(w)dw of a statistical model.
Hd ), the function K(g(u)) and the a priori distribution ϕ(g(u))|g (u)| are respectively expressed as K(g(u)) = u2k , ϕ(g(u))|g (u)| = φ(u)|uh |. The theorem that ensures the existence of such a real analytic manifold M and a real analytic map w = g(u) is called Hironaka’s theorem or resolution of singularities. The function w = g(u) is called a resolution map. In Chapters 2 and 3, we give a rigorous statement of the theorem and a method to find the set (M, g), respectively. 4 Four main formulas 29 we can prove that there exists a real analytic function a(x, u) such that f (x, g(u)) = a(x, u) uk (∀x).
Algebraic Geometry and Statistical Learning Theory by Sumio Watanabe