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

ISBN-10: 0521896193

ISBN-13: 9780521896191

This up to date account of algebraic information 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 figuring out of statistical versions, specifically, exponential versions. This e-book provides a brand new method of imminent classical statistical difficulties and increases clinical questions that will by no means were thought of with out the interplay of those disciplines. starting with a quick creation to every zone, utilizing uncomplicated illustrative examples, the e-book then proceeds with a set of studies and a few new effects written via top researchers of their respective fields. half III dwells in either classical and quantum details geometry, containing surveys of key effects and new fabric. ultimately, half IV offers examples of the interaction among algebraic records and knowledge geometry. machine code and proofs also are to be had on-line, the place key examples are built in additional element.

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

**Example text**

Pick any combination of r points along the hyper-surface S, say p(1) , . . e. the convex subset of ∆d−1 consisting of all points of the form h p(h) λh , for some choice of (λ1 , . . , λr ) ∈ ∆r −1 . 3). In fact, the closure of the union of all such convex hulls is precisely the latent class model H. In algebraic geometry, H would be described as the intersection of ∆d−1 with the r-th secant variety of the Segre embedding mentioned above. 32 S. E. Fienberg, P. Hersh, A. Rinaldo and Y. 1 The simplest example of a latent class model is for a 2 × 2 table with r = 2 latent classes.

The problem of characterising the singular locus of H and of computing the dimensions of its stratiﬁed components (and of the tangent spaces and tangent cones of its singular points) is of statistical importance: singularity points of H are probability distributions of lower complexity, in the sense that they are speciﬁed by lower-dimensional subsets of Θ, or, loosely speaking, by less parameters. Because the sample space is discrete, although the singular locus of H has typically Lebesgue measure zero, there is nonetheless a positive probability that the maximum likelihood estimates end up being either a Maximum likelihood estimation in latent class models 31 Fig.

6 Let R, S be commutative rings with identity. Then the set R ⊕ S = {(r, s) : r ∈ R, s ∈ S} with component-wise sum and product is a commutative ring with (1R , 1S ) as identity. 4 Let I, J be ideals in R such that I + J = R. Let φ : R → R/I ⊕ R/J be deﬁned as φ(f ) = ([f ]I , [f ]J ). It is an onto morphism, whose kernel is I ∩ J. Hence, R/(I ∩ J) is isomorphic to R/I ⊕ R/J. Localisation of a ring Let f ∈ R, f = 0, and let S = {f n : n ∈ N}. In R × S consider the equivalence relation (g, f m ) ∼ (h, f n ) if gf n = hf m .

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

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