By Sumio Watanabe

ISBN-10: 0521864674

ISBN-13: 9780521864671

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

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

**Example text**

From the definition of the Kullback–Leibler distance, f (x, g(u))q(x)dx = K(g(u)) = u2k . It follows that a(x, u)q(x)dx = uk . Moreover, by f (x, g(u)) = log(q(x)/p(x|g(u))), K(g(u)) = (f (x, g(u)) + e−f (x,g(u)) − 1)q(x)dx. It is easy to show t + e−t − 1 → 12 . 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).

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.

N→∞ (3) It is said that Xn converges to X in probability, if lim P (D(Xn , X) > ) = 0 n→∞ for arbitrary > 0, where D(·, ·) is the metric of the image space of X. 21 There are well-known properties of random variables. (1) If Xn converges to X almost surely or in the mean of order p > 0, then it does in probability. (2) If Xn converges to X in probability, then it does in law. For the definition of convergence in law, see chapter 5. 6 Probability theory 47 where X1 , . . , Xn are independently subject to the same distribution as X.

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

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