By Sumio Watanabe
Absolute to be influential, Watanabe's booklet lays the principles 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 the following underpins actual estimation strategies within the presence of singularities.
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Extra info for Algebraic Geometry and Statistical Learning Theory
17) i=1 Note that, in singular statistical models, sometimes inf Rn (w) = −∞, w which means that there is no parameter that minimizes Rn (w). If a parameter wˆ that minimizes Rn (w) exists, then a statistical estimation method ˆ Dn → p(x|w) is defined. The generalization error Rg and the training error Rt of this method are respectively defined by Rg = Rt = q(x) log 1 n n log i=1 q(x) dx, p(x|w) ˆ q(Xi ) . 3) respectively, they can be rewritten as ˆ Rg = K(w), ˆ Rt = Kn (w). 9 (Maximum likelihood and maximum a posteriori) (1) If an = 0 for arbitrary n, then wˆ is called the maximum likelihood (or ML) estimator and the statistical estimation method is called the maximum likelihood (or ML) method.
The partial expectation is defined by E[X]S = X(ω)P (dω) = X(ω)∈S x PX (dx). 20 These are fundamental remarks. 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].
23), we obtain the asymptotic expansion of v(t) for t → 0, ∞ mk v(t) = ckm t λk −1 (− log t)m−1 . 4 Four main formulas 33 holds for t → 0. Let Yn (w)dw be a measure defined by Yn (w)dw ≡ exp(−nβKn (w)) ϕ(w) dw, then we have an asymptotic expansion, Yn (w)dw = Yn (g(u)) |g (u)| du e−nβu = 2k √ + nβuk ξn (u) φ ∗ (u)|uh |du α ∞ = dt δ(t − u2k ) 0 α √ × φ ∗ (u)|uh |e−nβt+ ntβ ξn (u) du ∞ mk −1 = Dkm (u)du α k=1 r=0 ∞ × 0 dt n t n λk −1 log n t r √ e−βt+ tβ ξn (u) . 22). Such local coordinates are called essential coordinates in this book.
Algebraic Geometry and Statistical Learning Theory by Sumio Watanabe