This article focuses on the study of toric algebraic statistical models which
correspond to toric Del Pezzo surfaces with Du Val singularities. A closed-form
for the Maximum Likelihood Estimate of algebraic statistical models which
correspond to cubic and quartic toric Del Pezzo surfaces with Du Val singular
points is given. Also, we calculate the ML degrees of some toric Del Pezzo
surfaces of degree less than or equal to six, which equals the degree of the
surface in all the case but one, namely the quintic with two points of type
$\mathbb{A}_1$.Comment: 11 pages, 5 figure

Kosta, Dimitra

arXiv

In this article a closed-form for the Maximum Likelihood Estimate of algebraic statistical models which correspond to cubic and quartic toric del Pezzo surfaces with Du Val singular points is given

Edinburgh Research Explorer

     Edinburgh Research Explorer                                      A note on Maximum Likelihood Estimation for cubic and quarticcanonical toric del Pezzo SurfacesCitation for published version:Kosta, D 2016 'A note on Maximum Likelihood Estimation for cubic and quartic canonical toric del PezzoSurfaces' ArXiv.Link:Link to publication record in Edinburgh Research ExplorerDocument Version:Early version, also known as pre-printGeneral rightsCopyright for the publications made accessible via the Edinburgh Research Explorer is retained by the author(s)and / or other copyright owners and it is a condition of accessing these publications that users recognise andabide by the legal requirements associated with these rights.Take down policyThe University of Edinburgh has made every reasonable effort to ensure that Edinburgh Research Explorercontent complies with UK legislation. If you believe that the public display of this file breaches copyright pleasecontact openaccess@ed.ac.uk providing details, and we will remove access to the work immediately andinvestigate your claim.Download date: 05. Apr. 2019arXiv:1602.08307v1  [math.ST]  26 Feb 2016 A note on Maximum Likelihood Estimation forcubic and quartic canonical toric del PezzoSurfacesDimitra KostaFebruary 29, 2016AbstractIn this article a closed-form for the Maximum Likelihood Estimate ofalgebraic statistical models which correspond to cubic and quartic toricdel Pezzo surfaces with Du Val singular points is given.1 IntroductionMaximum likelihood estimation (MLE) is a standard approach to parameterestimation and inference, and a fundamental computational task in statistics.It consists of the following problem: given the observed data and a model ofinterest, find the probability distribution that is most likely to have producedthe data. In the past decade, algebraic techniques for the computation of max-imum likelihood estimates have been developed with some success for algebraicstatistical models for discrete data (see [1], [3], [4]).This article is concerned with the problem of Maximum Likelihood Estima-tion for algebraic statistical models with singularities, in particular those whichcorrespond to toric del Pezzo surfaces with Du Val singular points. The impor-tance of algebraic statistical models corresponding to toric varieties is due totheir relation to log-linear statistical models which are widely used in Statistics([2]). Another reason for studying the MLE for such algebraic statistical models,is that they correspond to singular varieties. Singularities play an importantrole in statistical inference as the commonly assumed smoothness of algebraicstatistical models is very restrictive and is almost never satisfied for models ofstatistical relevance (see [[2], p. 100], [8]).The relevant definitions to this problem are given bellow.The author would like to thank Ivan Cheltsov, Alexander Davie and Milena Hering forvaluable comments and corrections.11.1 Maximum Likelihood EstimationConsider the complex projective space Pn with coordinates (p0, p1, ..., pn). Inour setting the coordinate pi represents the probability of the i-event thereforep0 + p1 + ... + pn = 1. The set of points in Pn with positive real coefficients isidentified with the probability simplex∆n = {(p0, p1, ..., pn) ∈ Rn+1 : p0, p1, ..., pn ≥ 0 and p0 + p1 + ...+ pn = 1} .An algebraic statistical model is a closed subset M of the complex projectivespace Pn, with the model itself being the intersection ofM with the probabilitysimplex ∆n. The data is given by a non-negative integer vector (u0, u1, ..., un) ∈Nn+1, where ui is the number of times the i-event is observed.The maximul likelihood estimation problem consists of finding a model pointp which maximises the likelihood of observing the data. This amounts to max-imising the corresponding Likelihood FunctionL(p0, p1, ..., pn) =pu00 · pu11 · ... · punn(p0 + p1 + ...+ pn)(u0+u1+...+un)over the model M, where here we ignore a multinomial coefficient. Statisticalcomputations are usually implemented in the affine n-plane p0+p1+...+pn = 1.However, including the denominator makes the likelihood function a well-definedrational function on the projective space Pn, enabling one to use projectivealgebraic geometry to study its restriction to the variety M.The likelihood function might not be convex, so it can have many localmaxima and the problem of finding and certifying a global maximum is diffi-cult. Therefore, in most recent works the problem of finding all critical points ofthe likelihood function is considered, with the aim of identifying all local max-ima (see [1], [3] and [4]). This corresponds to solving a system of polynomialequations and these equations, defining the critical points of the likelihood func-tion L, are called likelihood equations. The number of complex solutions to thelikelihood equations equals the number of complex critical points of the restric-tion of the likelihood function L to the model M, which is called the MaximumLikelihood (ML) degree of the variety M.1.2 Toric modelsIn this article we are studying the Maximum Likelihood Estimation problem fortoric models which are models with a well behaved likelihood function. Toricmodels are known as log-linear models in statistics, because the logarithms of theprobabilities are linear functions in the logarithms of the parameters θi. Theyhave the property that maximum likelihood estimation is a convex optimizationproblem. Assuming that the parameter domain Θ is bounded, it follows that thelikelihood function has exactly one local maximum. We introduce toric modelsfollowing the notation used in Chapter 1.2 of [6].2Let A = (aij) be a non-negative integer d×m matrix with the property thatall column sums are equal:d∑i=1ai1 =d∑i=1ai2 = ... =d∑i=1aim .The j-th column vector aj of the matrix A represents the monomialθaj := θa1j1 · θa2j2 · · · θadjd for all j = 1, ...m.and the assumption that the column sums of the matrix A are all equal meansthese monomials all have the same degree.Definition 1.1 The toric model of A is the image of the orthant Θ = Rd>0under the mapf : Rd → Rm, θ 7→1∑mj=1 θaj· (θa1 , θa2 , ..., θam) .Maximum likelihood estimation for the toric model means solving the opti-mization problem of maximising the functionpu11 · pu21 · ... · pummsubject to the constrains f(Rd>0). This is equivallent to maximising functionθA·u subject to θ ∈ Rd>0 andm∑j=1θaj = 1,whereθAu =d∏i=1θai1u1+ai2u2+...+aimumi and θaj =d∏i=1θaiji .Let b := Au denote the sufficient statistic, then the optimisation problem abovebecomesMaximise θb subject to θ ∈ Rd>0 andm∑j=1θaj = 1 .Proposition 1.2 Fix a toric model A and data u ∈ Nm with sample size N =u1 + + um and sufficient statistic b = Au. Let pˆ = f(θˆ) be any local maximumfor the equivalent optimization problems above. ThenA · pˆ =1N· b.Given a matrix A ∈ Nd×m and any vector b ∈ Rd, we consider the setPA(b) = {p ∈ Rm : A · p =1N· b and pj > 0 for all j}.This is a relatively open polytope and Birch’s theorem below asserts that it iseither empty or meets the toric model in precisely one point.3Theorem 1.3 (Birch’s Theorem) Fix a toric model A and let u ∈ Nm>0 be astrictly positive data vector with sufficient statistic b = Au. The intersection ofthe polytope PA(b) with the toric model f(Rd>0) consists of precisely one point.That point is the maximum likelihood estimate pˆ for the data u.In the next section we will determine a closed form for the Maximum Like-lihood Estimates for all algebraic statistical models corresponding to toric delPezzo surfaces of degree three and four with Du Val singularities. The cor-responing polytope of a toric del Pezzo surface with Du Val singularities is areflexive polytope. According to the classification results of reflexive polytopes(see [5], [7]), there are 16 isomorphism classes of such two-dimensional reflexivepolytopes, a list of which we provide bellow.Proposition 1.4 There are exactly 16 isomorphism classes of two-dimensionalreflexive polytopes given in the list bellow. The number in the labels is the numberof lattice points on the boundary.42 Main CalculationIn this section we determine a closed form for the Maximum Likelihood estimatesfor all algebraic statistical models corresponding to cubic and quartic toric delPezzo surfaces with Du Val singularities.2.1 Cubic del Pezzo with three singular points of type A2Consider the case of a reflexive polytope with three lattice points on the bound-ary, as in the graph below. This corresponds to a cubic surface with three DuVal singular points of type A2.This polytope generated by the lattice points (1, 0), (0, 1), (−1,−1) in Z2gives the same toric variety as the polytope which is generated by the latticepoints (2, 1, 0), (1, 2, 0), (0, 0, 3) in Z3.We are interested in the algebraic statistical model given by the matrixA =2 1 01 2 00 0 3 .This non-negative integer matrix A corresponds to a toric algebraic varietywhich is the topological closure f(C3) of the image f(C3) ⊂ C3 under the mapf : C3 → C3 , (θ1, θ2, θ3) 7→1(θ21θ2 + θ1θ22 + θ33)(θ21θ2, θ1θ22 , θ33) .According to Birch’s theorem there is a unique maximum likelihood estimatepˆ for the data u = (u1, u2, u3) with N = u1+u2+u3. This unique MLE satisfiesthe equationA · pˆ =1N· A · u .Since det(A) 6= 0, the inverse A−1 exists and by multiplying the above matrixequation with A−1 from the left we getpˆ =1N· u .5This gives us the equationsθˆ21 θˆ2 =1Nu1 ,θˆ1θˆ22 =1Nu2 ,θˆ33 =1Nu3.and we can compute (θˆ1, θˆ2, θˆ3) = (3√u21N ·u2,3√u22N ·u1, 3√u3N).2.2 Quartic del Pezzo with four singular points of type A1.Again consider the case of a reflexive polytope with four lattice points on theboundary, as in the graph below.Again the polytope generated by the lattice points (1, 0), (0, 1), (−1, 0), (0,−1)in Z2 gives rise to the same toric variety as the polytope generated by the latticepoints (2, 1, 0), (1, 2, 0), (1, 0, 2), (0, 1, 2) in Z3.The non-negative integer matrixA =2 1 1 01 2 0 10 0 2 2corresponds to a toric algebraic variety which is the topological closure f(C3)of the image f(C3) ⊂ C4 under the mapf : C3 → C4 , (θ1, θ2, θ3) 7→1(θ21θ2 + θ1θ22 + θ1θ23 + θ2θ23)(θ21θ2, θ1θ22, θ1θ23 , θ2θ23) .According to Birch’s theorem there is a unique maximum likelihood estimateθˆ for the data u = (u1, u2, u3, u4) which satisfies the equationA · pˆ =1N· A · u ,6where N = u1+u2+u3+u4 and pˆ is the probability distribution correspondingto the parameter values (θˆ1, θˆ2, θˆ3). This gives us the equationsθˆ21 θˆ2 − θˆ2θˆ23 =1N(u1 − u4)θˆ1θˆ22 + θˆ2θˆ23 =1N(u2 + u4)θˆ1θˆ23 + θˆ2θˆ23 =1N(u3 + u4) ,and these equations give uspˆ1 =(u1 + u2)(u1 + u3)N2pˆ2 =(u1 + u2)(u2 + u4)N2pˆ3 =(u1 + u3)(u3 + u4)N2pˆ4 =(u2 + u4)(u3 + u4)N2.We can compute the unique maximum likelihood estimate θˆ for the data u, whichis (θˆ1, θˆ2, θˆ3) = (3√(u1+u2)(u1+u3)2N2(u2+u4), 3√(u1+u2)(u2+u4)2N2(u1+u3), 6√(u1+u3)(u2+u4)(u3+u4)3N4(u1+u2)).2.3 Quartic del Pezzo with one A2 and two A1 type singu-lar pointsAgain consider the case of a reflexive polytope with four lattice points on theboundary, as in the graph below.The polytope generated by the lattice points (1, 0), (0, 1), (−1, 1), (0,−1) in Z2 isthe same as the polytope generated by the lattice points (2, 1, 0), (1, 2, 0), (1, 0, 2), (0, 2, 1)in Z3.The non-negative integer matrixA =1 0 1 20 2 2 12 1 0 07corresponds to a toric algebraic variety which is the topological closure f(C3)of the image f(C3) ⊂ C4 under the mapf : C3 → C4 , (θ1, θ2, θ3) 7→1(θ1θ23 + θ22θ3 + θ1θ22 + θ21θ2)(θ1θ23 , θ22θ3, θ1θ22 , θ21θ2) .According to Birch’s theorem there is a unique maximum likelihood estimateθˆ for the data u = (u1, u2, u3, u4) which satisfies the equationA · pˆ =1N· A · u ,where N = u1+u2+u3+u4 and pˆ is the probability distribution correspondingto the parameter values (θˆ1, θˆ2, θˆ3).This gives us the equations2θˆ1θˆ23 + θˆ21 θˆ2 =1N(2u1 + u4)θˆ22 θˆ3 − θˆ21 θˆ2 =1N(u2 − u4)2θˆ1θˆ22 + 3θˆ21 θˆ2 =1N(2u3 + 3u4) .Then the unique maximum likelihood estimate θˆ for the data u is (θˆ1, θˆ2, θˆ3) =( 3√p24p3, 3√p23p4, 6√p33p24[p3 +3u1−u3N]). The probability distribution pˆ3 and pˆ4 aregiven by the quartic equationsaip4i + bip3i + cip2i + dipi + ei = 0 for i = 3, 4wherea4 = 19N4b4 = 2N3(19(u2 − u4)− 27N)c4 = 9N2(2u3 + 3u4)(6N − 6(u2 − u4)− 2(2u3 + 3u4))− 8N2(u2 − u4)2d4 = N(2u3 + 3u4)2[8(2u3 + 3u4)− 9(2N − 2(u2 − u4))]e4 = (2u3 + 3u4)3[2N − 2(u2 − u4)− (2u3 + 3u4)] ,anda3 = 11N4b3 = 48N4 − 11N3(−3u1 + u3)c3 = 36N2(u2 − u4)2 − 8N2(3u2 + 2u3)(2u3 + 3u4)d3 = −4N3(3N − 3u1 + u3)e4 = (3u2 + 2u3)2(2u3 + 3u4)2 .82.4 Quartic Del Pezzo surface with one A3 and two A1 typesingular points.Again consider the case of a reflexive polytope with four lattice points on theboundary, as in the graph below.We once more we can see that the polytope generated by the lattice points(1, 1), (0, 1), (−1, 1), (0,−1) in Z2 gives the same toric variety as the polytopegenerated by the lattice points (1, 0, 3), (2, 2, 0), (1, 2, 1), (0, 2, 2) in Z3.The non-negative integer matrixA =1 2 1 00 2 2 23 0 1 2corresponds to a toric algebraic variety which is the topological closure f(C3)of the image f(C3) ⊂ C4 under the mapf : C3 → C4 , (θ1, θ2, θ3) 7→1(θ1θ33 + θ21θ22 + θ1θ22θ3 + θ22θ23)(θ1θ33, θ21θ22, θ1θ22θ3, θ22θ23) .According to Birch’s theorem there is a unique maximum likelihood estimateθˆ for the data u = (u1, u2, u3, u4) which satisfies the equationA · pˆ =1N· A · u ,where N = u1+u2+u3+u4 and pˆ is the probability distribution correspondingto the parameter values (θˆ1, θˆ2, θˆ3).This gives us the equationsθˆ1θˆ33 =1Nu1θˆ21 θˆ22 − θˆ22 θˆ23 =1N(u2 − u4)θˆ1θˆ22 θˆ3 + 2θˆ22 θˆ23 =1N(u3 + 2u4) .and these equations give uspˆ1 =u1N9We can compute the unique maximum likelihood estimate θˆ for the data u,which isθˆ1 =4√√√√u1(u2 − u4)38N( 2√(u2−u4)2+4(u3+2u2)(u3+2u4)(u2−u4)2− 1)3(u3 + 2u2)3θˆ2 =4√(u2 + u3 + u4)(u3 + 2u4)3Nu1(u2 + 2u3 + 3u4)2θˆ3 =4√√√√ u1(u2 − u4)2N(u3 + 2u4)(1 + 2√(u2 − u4)2 + 4(u3 + 2u2)(u3 + 2u4)(u2 − u4)2)2.5 Toric Del Pezzo surfaces of degree greater than five.When the degree of the Del Pezzo surface is greater than five, the definingequations of the probability distribution (pˆ1, pˆ2, pˆ3, pˆ4) have degree 5 and wecannot obtain a closed form for the Maximum Likelihood Estimate. Although aclosed-form formula for maximum likelihood estimates is not achieved for theselog-linear models, the log-likelihood function is convex for these models, andany hill-climbing algorithm can be used to compute the ML estimates.References[1] F. Catanese, S. Hosten, A.Khetan and B. Sturmfels, The Maximum likeli-hood degree, American Journal of Mathematics, 18 (2006), 671-697.[2] M.Drton, B. Sturmfels, S. Sullivant, Lectures on Algebraic Statistics, Ober-wolfach Seminars, 39 (2008), Birkhuser, Basel.[3] J. D.Hauenstein, J. Rodriguez, and B. Sturmfels, Maximum likelihood formatrices with rank constraints, Journal of Algebraic Statistics, 5 (2014),18-38.[4] S. Hosten, A.Khetan and B. Sturmfels, Solving the likelihood equations,Foundations of Computational Mathematics, 5 (2005), 389-407.[5] M.Kreuzer, H. Skarke, On the classification of reflexive polyhedra, Com-mun. Math. Phys. 185 (1997) 495-508.[6] L. Pachter and B. Sturmfels, Algebraic Statistics for Computational Biology,(2005) Cambridge University Press, New York.[7] H. Sato, Toward the classification of higher-dimensional Fano varieties, To-hoku Math. J., 52 (2000), 383-413.10[8] B. Sturmfels, Open problems in Algebraic Statistics, In ”Emerging Appli-cations of Algebraic Geometry”, I.M.A. Volumes in Mathematics and itsApplications, 149 (2009), Springer, New York, 351-364.Dimitra KostaSchool of Mathematics,The University of Edinburgh,Edinburgh EH9 3JZ, UK.D.Kosta@ed.ac.uk11

A note on Maximum Likelihood Estimation for cubic and quartic canonical toric del Pezzo Surfaces

Abstract

Similar works

Full text

Available Versions

Edinburgh Research Explorer