By Sumio Watanabe

ISBN-10: 0521864674

ISBN-13: 9780521864671

Guaranteed to be influential, Watanabe's publication lays the rules for using algebraic geometry in statistical studying concept. Many models/machines are singular: mix types, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the speculation accomplished right here underpins actual estimation suggestions within the presence of singularities.

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**Additional resources for Algebraic Geometry and Statistical Learning Theory**

**Example text**

Max max{0, ξ (v)} α Hence the generalization and training errors are given by Rg = 1 max {0, ξ (u)}2 , 4n u∈M0 Rt = − 1 max {0, ξ (u)}2 , 4n u∈M0 where M0 = g −1 (W0 ) is the set of true parameters. The symmetry of generalization and training errors holds if an /np → ∞ for arbitrary p > 0. Therefore, E[nRg ] = −E[nRt ] + o(1). For the other sequence an , the same result is obtained. Main Formula IV (Symmetry of generalization and training errors) If the maximum likelihood or generalized maximum a posteriori method is applied, the symmetry of generalization and training errors holds, lim E[nRg ] = − lim E[nRt ].

4) In the set A = {(x, y); y 2 − x 3 = 0}, the origin is a singularity of A, which is a critical point of the function f (x, y) = y 3 − x 2 . (5) In the set A = {(x, y); x 5 − y 3 = 0}, the origin is a singularity of A, which is a critical point of f (x, y) = x 5 − y 3 . The set A has a tangent line y = 0. (6) In the set A = {(x, y, z); xyz = 0}, Sing(A) = {(x, y, z); x = y = 0, or y = z = 0, or z = x = 0}. The set B = {(x, y, z); x = y = 0} is a nonsingular set contained in Sing(A). Such a set is called a nonsingular set contained in the singular locus of A.

The expectation of f (X) is equal to E[f (X)] = f (X(ω))P (dω) = f (x) PX (dx). This expectation is often denoted by EX [f (X)]. (2) Two random variables which have the same probability distribution have the same expectation value. Hence if X and Y have the same probability distribution, we can predict E[Y ] based on the information of E[X]. (3) In statistical learning theory, it is important to predict the expectation value of the generalization error from the training error. (4) If E[|X|] = C then, for arbitrary M > 0, C = E[|X|] ≥ E[|X|]{|X|>M} ≥ ME[1]{|X|>M} = MP (|X| > M).