By Sumio Watanabe

ISBN-10: 0521864674

ISBN-13: 9780521864671

Absolute to be influential, Watanabe's booklet lays the principles for using algebraic geometry in statistical studying thought. Many models/machines are singular: combination versions, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the idea completed the following underpins exact estimation concepts within the presence of singularities.

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

**Example text**

4 Four main formulas 39 where Rg and Rt are the generalization and training errors of the maximum likelihood estimator. 34) is well known as the Akaike information criterion (AIC) of a regular statistical model, hence Main Formula III contains AIC as a very special case. 33) does not hold in general, hence AIC cannot be applied. Moreover, Main Formula III holds even if the true distribution is not contained in the model [120]. 4 ML and MAP theory The last formula concerns the maximum likelihood or a posteriori method.

Here we attain the first main formula. 19) where ξn (u) converges in law to the Gaussian process ξ (u). 20) where φ(u) > 0 is a positive real analytic function. 14 (1) Note that the log likelihood ratio function of any singular statistical model can be changed to the standard form by algebraic geometrical transform, which allows |g (u)| = 0. (2) The integration over the manifold M can be written as the finite sum of the integrations over local coordinates. There exists a set of functions {σα (u)} such that σα (u) ≥ 0, α σα (u) = 1, and the support of σα (u) is contained in Mα .

However, in U = {(x, y) ∈ R2 ; 0 < x, y < 1}, {(x, y) ∈ U ; f (x, y) = 0} = {(x, y) ∈ U ; xy = 1/(nπ ), n = 1, 2, . } is a real analytic set. 3 Singularity Let U be an open set in Rd and f : U → R1 be a function of C 1 class. The d-dimensional vector ∇f (x) ∈ Rd defined by ∇f (x) = ∂f ∂f ∂f (x), (x), . . , (x) ∂x1 ∂x2 ∂xd is said to be the gradient vector of f (x). 4 (Critical point of a function) Let U be an open set of Rd , and f : U → R1 be a function of C 1 class. (1) A point x ∗ ∈ U is called a critical point of f if it satisfies ∇f (x ∗ ) = 0.

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

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