By Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, Henry P. Wynn

ISBN-10: 0521896193

ISBN-13: 9780521896191

This updated account of algebraic facts and data geometry explores the rising connections among the 2 disciplines, demonstrating how they are often utilized in layout of experiments and the way they profit our knowing of statistical types, specifically, exponential versions. This publication offers a brand new method of impending classical statistical difficulties and increases medical questions that may by no means were thought of with out the interplay of those disciplines. starting with a short advent to every region, utilizing uncomplicated illustrative examples, the e-book then proceeds with a set of reports and a few new effects written by means of best researchers of their respective fields. half III dwells in either classical and quantum info geometry, containing surveys of key effects and new fabric. eventually, half IV offers examples of the interaction among algebraic facts and data geometry. computing device code and proofs also are to be had on-line, the place key examples are constructed in additional aspect.

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**Extra resources for Algebraic and Geometric Methods in Statistics**

**Sample text**

Fienberg, P. Hersh, A. Rinaldo and Y. Zhou Fig. 12), when evaluated over the ball in R3 of radius 3, centred at the origin. The inner box is the unit cube [0, 1]3 .

Xak k for a = (a1 , . . , ak ) ∈ (Z≥0 )k . The set of terms is indicated as Tk . The operation in Tk , of interest, is the product of terms. 15 A term-ordering is a well ordering on Tk such that 1 every xa ∈ Tk and xa xb implies xa xc xb xc for every xc ∈ Tk . xa for A polynomial in R is a linear combination of a ﬁnite set of terms in Tk : f = a k a∈A ca x where A is a ﬁnite subset of Z≥0 . 16 Let f ∈ R be a polynomial, A the ﬁnite set formed by the terms in f and xb = max {xa : a ∈ A}. Let I ⊂ R be an ideal.

Ft ∈ I. The set {f1 , . . , ft } is a Gr¨ obner basis of I with respect to if LT(I) = LT(f1 ), . . , LT(ft ) . Gr¨ obner bases are special sets of generators for ideals in R. Among the many results concerning Gr¨ obner bases, we list a few, to stress their role in the theory of ideals in polynomial rings. 10 Let I ⊆ R be an ideal. Then, I = R if, and only if, 1 ∈ F, where F is a Gr¨ obner basis of I, with respect to any term-ordering . 11 Let I ⊂ R be an ideal. The ring R/I is 0–dimensional if, and only if, xai i ∈ LT(I) for every i = 1, .

### Algebraic and Geometric Methods in Statistics by Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, Henry P. Wynn

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