By Sumio Watanabe

ISBN-10: 0521864674

ISBN-13: 9780521864671

Guaranteed to be influential, Watanabe's ebook lays the principles for using algebraic geometry in statistical studying conception. Many models/machines are singular: blend versions, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the speculation accomplished the following underpins actual estimation ideas within the presence of singularities.

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**Extra info for Algebraic Geometry and Statistical Learning Theory**

**Example text**

14 and ( 2 , B2 ) be a measurable space. If f : 1 → 2 is a measurable function then f (X) is a random variable on ( , B, P ). 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.

T→0 t2 lim Therefore, if u2k = 0, then a(x, u)2 q(x)dx = lim u2k →0 2K(g(u)) = 2. u2k Here we can introduce a well-defined stochastic process on M, 1 ξn (u) = √ n n {uk − a(Xi , u)}, i=1 from which we obtain a representation, nKn (g(u)) = nu2k − √ k nu ξn (u). 18) By definition, ξn (u) satisfies E[ξn (u)] = 0 (∀u ∈ M), E[ξn (u)ξn (v)] = EX [a(X, u)a(X, v)] − uk v k (∀u, v ∈ M). If K(g(u)) = K(g(v)) = 0, then E[ξn (u)ξn (v)] = EX [a(X, u)a(X, v)], and E[ξn (u)2 ] = 2. By the central limit theorem, for each u ∈ M, ξn (u) converges in law to a Gaussian distribution with mean zero and variance 2.

If K(w) is a polynomial, then W0 is called an algebraic set; if K(w) is an analytic function, then W0 is called an analytic set. If W0 is not one point, neither the Bayes a posteriori distribution nor the distribution of the maximum likelihood estimators converges to the normal distribution. 5 shows a Bayes a posteriori distribution when the set of true parameters is {(a, b); ab = 0}. The right-hand side shows the probability distribution of the maximum likelihood estimator. We need a method to analyze such a singular distribution.

### Algebraic Geometry and Statistical Learning Theory by Sumio Watanabe

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