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University of California – Berkeley

1. Lin, Shaowei. Algebraic methods for evaluating integrals In Bayesian statistics.

Degree: Mathematics, 2011, University of California – Berkeley

URL: http://www.escholarship.org/uc/item/6r99035v

The accurate evaluation of marginal likelihood integrals is a difficult fundamental problem in Bayesian inference that has important applications in machine learning and computational biology. Following the recent success of algebraic statistics in frequentist inference and inspired by Watanabe's foundational approach to singular learning theory, the goal of this dissertation is to study algebraic, geometric and combinatorial methods for computing Bayesian integrals effectively, and to explore the rich mathematical theories that arise in this connection between statistics and algebraic geometry. For these integrals, we investigate their exact evaluation for small samples and their asymptotics for large samples.According to Watanabe, the key to understanding singular models lies in desingularizing the Kullback-Leibler function K(w) of the model at the true distribution. This step puts the model in a standard form so that various central limit theorems can be applied. While general algorithms exist for desingularizing any analytic function, applying them to non-polynomial functions such as K(w) can be computationally expensive. Many singular models are however represented as regular models whose parameters are polynomial functions of new parameters. Discrete models and multivariate Gaussian models are all examples. We call them regularly parametrized models. One of our main contributions is showing how this polynomiality can be exploited by defining fiber ideals for singular models and relating the properties of these algebraic objects to the statistics. In particular, we prove that a model is put in standard form if we monomialize the corresponding fiber ideal. As a corollary, the learning coefficient of a model is equal to the real log canonical threshold (RLCT) of the fiber ideal.While complex log canonical thresholds are well-studied in algebraic geometry, little is known about their real analogs. In Chapter 4, we prove their fundamental properties and simple rules of computation. We also extend Varchenko's notion of Newton polyhedra and nondegeneracy for functions to ideals. Using these methods, we discover a formula for the RLCT of a monomial ideal with respect to a monomial amplitude. For all other ideals, this formula is an upper bound for their RLCT. Our tools are then applied to a difficult statistical example involving a naive Bayesian network with two ternary random variables.Because our statistical models are defined over compact semianalytic parameter spaces W, we need to extend standard asymptotic theory of real analytic functions over neighborhoods of the origin to functions over domains like W. Chapter 3 summarizes these results which are critical for other proofs in this dissertation. We also give explicit formulas for the full asymptotic expansion of a Laplace integral over W in terms of the Laurent coefficients of the associated zeta function. In Chapter 5, we apply these formulas to Laplace integrals Z(n) with nondegenerate phase functions, and describe algorithms for computing the coefficient C in…

Subjects/Keywords: Mathematics; Applied Mathematics; Statistics; Algebraic Geometry; Asymptotics; Bayesian Integral; Real Log Canonical Threshold; Resolution of Singularities; Singular Learning Theory

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Lin, S. (2011). Algebraic methods for evaluating integrals In Bayesian statistics. (Thesis). University of California – Berkeley. Retrieved from http://www.escholarship.org/uc/item/6r99035v

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Lin, Shaowei. “Algebraic methods for evaluating integrals In Bayesian statistics.” 2011. Thesis, University of California – Berkeley. Accessed July 12, 2020. http://www.escholarship.org/uc/item/6r99035v.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Lin, Shaowei. “Algebraic methods for evaluating integrals In Bayesian statistics.” 2011. Web. 12 Jul 2020.

Vancouver:

Lin S. Algebraic methods for evaluating integrals In Bayesian statistics. [Internet] [Thesis]. University of California – Berkeley; 2011. [cited 2020 Jul 12]. Available from: http://www.escholarship.org/uc/item/6r99035v.

Note: this citation may be lacking information needed for this citation format:

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Lin S. Algebraic methods for evaluating integrals In Bayesian statistics. [Thesis]. University of California – Berkeley; 2011. Available from: http://www.escholarship.org/uc/item/6r99035v

Not specified: Masters Thesis or Doctoral Dissertation

University of Illinois – Chicago

2.
Song, Lei.
Rational Singularities of Brill-Noether Loci and *Log* *Canonical* Thresholds on Hilbert Schemes of Points.

Degree: 2014, University of Illinois – Chicago

URL: http://hdl.handle.net/10027/18980

It is well known in algebraic geometry that Hilbert and Picard functors are representable by Hilbert schemes {Hilb}(X) and Picard schemes {Pic}(X) respectively. The thesis studies singularities of certain spaces relating to these schemes. It primarily consists of two parts of independent interest.
In the first part (Chapter 3), we study the Brill-Noether locus W^{0}(X) of effective line bundles over a smooth projective variety X of arbitrary dimension; and we show that if a line bundle L is semi-regular, then W^{0}(X) has rational singularities at [L]. Since the semi-regularity holds automatically for all line bundles over a curve, we thereby recover a Kempf's theorem stating that all Brill-Noether loci W^{0}_{d}(C) have rational singularities for all smooth projective curve C of genus g and 1 ≤ d ≤ g-1. We also study the local ring \sshf{W^{0}(X), [L]} for such L. To show the condition of semi-regularity is not overly strong, we construct a family of examples from ruled surfaces, and make an analysis of one type of components of W^{0}_{sr}(X).
In the second part (Chapter 4), we study the Hilbert scheme of n-points on a quasi-projective smooth surface X. Specifically, we show that the universal family Z^{n} over {Hilb}^{n}(X) has non ℚ-Gorenstein, rational singularities, and its Samuel multiplicity can be described by a quadric in terms of the dimension of socle of zero-dimensional subscheme. In a different but closely related direction, we study the log canonical threshold c_{n} of the pair ({Hilb}^{n}(X), B^{n}), where X is the affine plane and B^{n} is the exceptional divisor of the Hilbert-Chow morphism, via two approaches. Using the Fulton-MacPherson compactification of configuration spaces and Haiman's work on the n! conjecture, we give a lower bound of c_{n}. On the other hand, by versal deformations of monomial ideals on the plane, we relate c_{n} to the log canonical threshold of the discriminant of a degree n polynomial in one variable.
*Advisors/Committee Members: Ein, Lawrence (advisor), Coskun, Izzet (committee member), Libgober, Anatoly (committee member), Popa, Mihnea (committee member), Niu, Wenbo (committee member).*

Subjects/Keywords: Brill-Noether loci; Semi-regular line bundles; Rational singularities; Hilbert scheme of points on a surface; Universal family; Log canonical threshold

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Song, L. (2014). Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. (Thesis). University of Illinois – Chicago. Retrieved from http://hdl.handle.net/10027/18980

Not specified: Masters Thesis or Doctoral Dissertation

Chicago Manual of Style (16^{th} Edition):

Song, Lei. “Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points.” 2014. Thesis, University of Illinois – Chicago. Accessed July 12, 2020. http://hdl.handle.net/10027/18980.

Not specified: Masters Thesis or Doctoral Dissertation

MLA Handbook (7^{th} Edition):

Song, Lei. “Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points.” 2014. Web. 12 Jul 2020.

Vancouver:

Song L. Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. [Internet] [Thesis]. University of Illinois – Chicago; 2014. [cited 2020 Jul 12]. Available from: http://hdl.handle.net/10027/18980.

Not specified: Masters Thesis or Doctoral Dissertation

Council of Science Editors:

Song L. Rational Singularities of Brill-Noether Loci and Log Canonical Thresholds on Hilbert Schemes of Points. [Thesis]. University of Illinois – Chicago; 2014. Available from: http://hdl.handle.net/10027/18980

Not specified: Masters Thesis or Doctoral Dissertation