By Sumio Watanabe

Certain to be influential, Watanabe's e-book lays the rules for using algebraic geometry in statistical studying thought. Many models/machines are singular: mix versions, neural networks, HMMs, Bayesian networks, stochastic context-free grammars are significant examples. the speculation completed right here underpins exact estimation strategies within the presence of singularities.

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**Example text**

The real Euclidean space Rd is a complete and separable metric space. (2) A subset of Rd is a metric space with the same metric. Sometimes a finite or countable subset in Rd is studied. (3) Let K be a compact subset in Rd . The set of all continuous function from K to Rd = {f ; f : K → Rd } is a metric space with the metric D(f, g) = f − g ≡ max |f (x) − g(x)|, x∈K where | · | is the norm of Rd . By the compactness of K in Rd , it is proved that is a complete and separable metric space. 12 (Probability space) Let be a metric space.

1) A point P in A is said to be nonsingular if there exist open sets U, V ⊂ Rd and an analytic isomorphism f : U → V such that f (A ∩ U ) = {(x1 , x2 , . . , xr , 0, 0, . . 3). If all points of A are nonsingular, then A is called a nonsingular set. 3 Singularity 55 (2) If a point P in A is not nonsingular, it is called a singularity or a singular point of the set A. The set of all singularities in A is called the singular locus of A, which is denoted by Sing(A) = {P ∈ A ; P is a singularity of A}.

4 (1) A function on R2 f (x, y) = x 2 + y 4 + 3 has a unique local minimum point (0, 0). (2) For a function on R3 f (x, y, z) = (x + y + z)4 + 1, all points (x ∗ , y ∗ , z∗ ) which satisfy x ∗ + y ∗ + z∗ = 0 are local minimum points. , xd U f V P A f (A) x1, x2 , ... , xr Fig. 3. Definition of a nonsingular point there is no local maximum or minimum point. The origin (0, 0) is a critical point of f . If |x| ≥ |y|, then f (x, y) ≥ 0, and if |x| ≤ |y|, then f (x, y) ≤ 0. Such a critical point is said to be a saddle point.