By Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, Henry P. Wynn
This up to date account of algebraic records 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 gain our knowing of statistical types, specifically, exponential types. This ebook provides a brand new method of imminent classical statistical difficulties and increases clinical questions that might by no means were thought of with out the interplay of those disciplines. starting with a short advent to every region, utilizing basic illustrative examples, the booklet then proceeds with a suite of studies and a few new effects written by way of major researchers of their respective fields. half III dwells in either classical and quantum info geometry, containing surveys of key effects and new fabric. ultimately, 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 built in additional aspect.
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Additional info for Algebraic and Geometric Methods in Statistics
16) applies to the simplex case and exhibits a relationship endowed with the Fisher information. Let u = log pq so that for the exponential model p˙λ = ∂pλ = u − Eλ [u]. ∂λ Now the mixture representative of the models is ppλ − 1, whose diﬀerential (in the tangent space) is puλ = pq v, say. 16) becomes u ¯, v¯ p ¯, v¯ = u p,F R = Covp (u, v). Note that the manifold Pn1 with the Fisher metric is isometric with an open subset of the sphere of radius 2 in Rn . , √ p1 pn . (Gibilisco and Isola 2001) shows that the Fisher information metric is the pull-back of the natural metric on the sphere.
1992, Ch. 3), to eliminate all the marginal probabilities and to study the Gr¨ obner basis of the resulting ideal in which the indeterminates are the joint probabilities only. There is only one element in the basis, namely p111 + p112 + p121 + p122 + p211 + p212 + p221 + p222 = 1, which gives the trivial condition for probability vectors. 6) is surjective, so that H = ∆7 and the eﬀective dimension is also 7, showing identiﬁability, at least for positive distributions. 3 We consider the 2 × 2 × 3 table with r = 2.
1. e. the lines connecting any two distinct points lying on the surface of independence. 1 displays ﬁve such secant lines. It is not to hard to picture that the union of all such secant lines is the enveloping simplex ∆3 and, therefore, H ﬁlls up all the available space. 3 in (Catalisano et al. 2002). The model H, thought of as a portion of the r-th secant variety to the Segre embedding, is not a smooth manifold. , (Benedetti 1990), clearly singular on the boundary of the simplex, but also at strictly positive points along the (r − 1)st secant variety (both of Lebesgue measure zero).
Algebraic and Geometric Methods in Statistics by Paolo Gibilisco, Eva Riccomagno, Maria Piera Rogantin, Henry P. Wynn