Toric Varieties in Statistics

Transcription

1 Toric Varieties in Statistics Daniel Irving Bernstein and Seth Sullivant North Carolina State University Daniel Irving Bernstein Toric Varieties in Statistics / 37

2 Outline Algebraic Statistics: Log-Linear Models Warm-up Example General Theory Hierarchical Models - Current Research What is a Hierarchical Model? Unimodularity Normality Daniel Irving Bernstein Toric Varieties in Statistics / 37

3 Outline Algebraic Statistics: Log-Linear Models Warm-up Example General Theory Hierarchical Models - Current Research What is a Hierarchical Model? Unimodularity Normality Daniel Irving Bernstein Toric Varieties in Statistics 3 / 37

4 Example Death Penalty Defendant s Race Yes No Total White Black Total Table: Homicide indictments in Florida in the 970s [4] Did race play a role in determining whether someone received the death penalty? Let p wy be the probability that a defendant is white, and is given the death penalty Define p wn, p by, p bn analogously Race and sentencing are independent iff p wy p bn p wn p by = 0 Daniel Irving Bernstein Toric Varieties in Statistics 4 / 37

5 Are they independent? Let S be the set of tables with the same row and column sums as data If p wy p bn p wn p by = 0 then Pr(U = u U S) = u wy!u wn!u by!u bn! v S v wy!v wn!v by!v bn! The χ test statistic for our data u should be small χ (U) = (U wy û wy ) û wy + (U wn û wn ) û wn + (U by û by ) û by + (U bn û bn ) where the ûs are the expected values Compute the p-value p = Pr(χ (U) χ (u) U S) If p value is small, then independence is unlikely û bn Daniel Irving Bernstein Toric Varieties in Statistics 5 / 37

6 Enumerating S S is set of nonnegative integer matrices with row and column sums ( ) 60 r = and c = ( ) 66 Each matrix in S is u + λm for some λ Z where ( ) m = u = ( 9 ) u + 6m = ( 5 ) Enumeration in most other models is infeasible, so use random sample Daniel Irving Bernstein Toric Varieties in Statistics 6 / 37

7 Log-Linear Models Definition A statistical model is a collection of probability distributions that satisfy some given conditions. Many statistical models can be defined algebraically. Definition Let A Z d n be an integer matrix. We define the toric ideal associated to A to be I A = x u x v u, v N n with Au = Av K[x,..., x n ]. We define the log-linear model associated to A to be M A = int( n ) V (I A ) where n = {p R n p i 0 and n i= p i = }. Daniel Irving Bernstein Toric Varieties in Statistics 7 / 37

8 Example The independence model can be realized as a log-linear model. 0 0 A = ker A = Span R 0 0 I A = x x 4 x x 3 M A = V (I A ) int( 3 ) Daniel Irving Bernstein Toric Varieties in Statistics 8 / 37

9 Hypothesis Testing for Log Linear Models u N n is data, randomly generated according to some (unknown) distribution p = (p,..., p n ) Null hypothesis: p M A In this case, distribution on {v N n Av = Au} is Test statistic Pr(U = u AU = Au) = χ (U) := where û i are the expected values / ( n i= u i!) v N n :Av=Au / ( n i= v i!) n (U i û i ) If p = Pr(χ (U) χ (u) AU = Au) is small, then our null hypothesis is probably wrong i= û i Daniel Irving Bernstein Toric Varieties in Statistics 9 / 37

10 Monte Carlo Method for Computing χ (u) Define F A,b := {v N n Av = b}. Usually finite for all b Create graph on F A,b Select {m,..., m k } ker Z A Edge between u, v F A,b iff u = v ± m i, some i Theorem (Diaconis-Sturmfels 998) If the graph on F A,b is connected, then a certain random walk on F produces a sequence v, v,... such that with probability, lim M M M {χ (v i ) χ (u)} = Pr(χ (U) χ (u) AU = Au) t= Theorem (Diaconis-Sturmfels 998) Let {m,..., m k } ker Z A. The graph on F A,b is connected for all b N d iff I A = x m+ i x m i i =,..., k. Daniel Irving Bernstein Toric Varieties in Statistics 0 / 37

11 Example A = m = m+ = 0 0 m = 0 0 b = F A,b = I A = x x 4 x x m m Daniel Irving Bernstein Toric Varieties in Statistics / 37

12 Markov Bases Definition We call M := {m,..., m k } ker Z A a Markov basis if I A = x m+ i x m i i =,..., k. Question Given A, can we efficiently compute a Markov basis? Daniel Irving Bernstein Toric Varieties in Statistics / 37

13 Outline Algebraic Statistics: Log-Linear Models Warm-up Example General Theory Hierarchical Models - Current Research What is a Hierarchical Model? Unimodularity Normality Daniel Irving Bernstein Toric Varieties in Statistics 3 / 37

14 Hierarchical Models Definition (Hierarchical Model) Let X,..., X n be discrete random variables. A simplicial complex C on X,..., X n specifies independence relations among the X i s. The collection of probability distributions on X,..., X n satisfying these relations is called a hierarchical model. X is independent of Y and Z, but Y and Z are dependent X Y Z There is no 3-way dependence Y X Z X and Z are independent of W given Y X Y Z W Daniel Irving Bernstein Toric Varieties in Statistics 4 / 37

15 Hierarchical Models as Log-Linear Models - Example Assume X,Y, and Z have 3,, and states. Independence relationship: X Y Z The matrix that maps 3 tables to the down and left and back margins realizes this as a log-linear model sum going down: front 0 3 ( ) back 3 0 sum going left and back: Daniel Irving Bernstein Toric Varieties in Statistics 5 / 37

16 Hierarchical Models as Log Linear Models Discrete random variables X,..., X n X i has d i states. Notation: d = (d,..., d n ) C denotes a simplicial complex on [n] The corresponding hierarchical model is a log-linear model with the following matrix Definition Let A C,d be the matrix defined as follows: Columns are indexed by elements of n i= [d i] Rows are indexed by F facet(c) j F [d j] Entry in row (F, (j,..., j k )) and column (i,..., i n ) is if i F = (j,..., j k ) All other entries are 0 Daniel Irving Bernstein Toric Varieties in Statistics 6 / 37

17 Example Let n = 3 with d = 3, d =, d 3 = Let C be the complex 3 Then A C,d is the following matrix: {}, {}, {}, {, 3}, {, 3}, {, 3}, {, 3}, Daniel Irving Bernstein Toric Varieties in Statistics 7 / 37

18 Unimodularity Definition (Unimodularity) Assume A Z d n has full row rank. We say that A is unimodular if all d d submatrices have determinant 0,, or. Example The matrix A is unimodular, whereas B is not ( ) ( ) 0 A = B = Applications include: Integer programming over fibers F A,b Disclosure limitation Computing Markov basis and universal Gröbner basis of I A Daniel Irving Bernstein Toric Varieties in Statistics 8 / 37

19 Unimodularity Question When is A C,d unimodular? Observation If A C,d is unimodular, then so is A C := A C,(,...,). Terminology abuse C is unimodular means A C is unimodular We have a complete classification of unimodular C Daniel Irving Bernstein Toric Varieties in Statistics 9 / 37

20 Unimodularity-Preserving Operations Definition (Adding a cone vertex) If C is a simplicial complex on [n], define cone(c) to be the complex on [n + ] with facets facet(cone(c)) = {F {n + } : F facet(c)}. Daniel Irving Bernstein Toric Varieties in Statistics 0 / 37

21 Unimodularity-Preserving Operations Definition (Adding a ghost vertex) If C is a simplicial complex on [n], define G(C) to be the simplicial complex on [n + ] that has exactly the same faces as C. Daniel Irving Bernstein Toric Varieties in Statistics / 37

22 Unimodularity-Preserving Operations Definition (Alexander Duality) If C is a simplicial complex on [n], then the Alexander dual complex C is the simplicial complex on [n] with facets facet(c ) = {[n] \ S : S is a minimal non-face of C}. Daniel Irving Bernstein Toric Varieties in Statistics / 37

23 Unimodularity: Constructive Classification Definition We say that a simplicial complex C is nuclear if it satisfies one of the following: C = k for some k (i.e. a simplex) C = m n (i.e. a disjoint union of simplices) 3 C = cone(d) where D is nuclear 4 C = G(D) where D is nuclear 5 C is the Alexander dual of a nuclear complex. Theorem (B.-Sullivant 05) The matrix A C is unimodular if and only if C is nuclear. Daniel Irving Bernstein Toric Varieties in Statistics 3 / 37

24 Simplicial Complex Minors Definition (Deletion and Link) Let C be a simplicial complex on [n]. Let v [n] be a vertex of C. Then C \ v denotes the induced simplicial complex on [n] \ {v}, and link v (C) denotes the simplicial complex on [n] \ {v} with facets facet(link v (C)) = {F \ {v} : F is a facet of C with v F }. Definition (Simplicial Complex Minor) We say that D is a minor of C if D can be obtained from C via a series of deletion and link operations. C = v C \ v = link v C = Daniel Irving Bernstein Toric Varieties in Statistics 4 / 37

25 Unimodularity: Excluded Minor Classification Theorem (B.-Sullivant 05) The matrix A C is unimodular if and only if C has no simplicial complex minors isomorphic to any of the following k {v}, the disjoint union of the boundary of a simplex and an isolated vertex O 6, the boundary complex of an octahedron, or its Alexander dual O 6 The four simplicial complexes shown below Daniel Irving Bernstein Toric Varieties in Statistics 5 / 37

26 Sketch of Proof C nuclear = C unimodular Simplices are unimodular A disjoint union of two simplices is unimodular Adding cone and ghost vertices and taking duals preserves unimodularity C unimodular = C avoids forbidden minors The forbidden minors are not unimodular Taking minors preserves unimodularity C avoids forbidden minors = C nuclear If C avoids the forbidden minors but has a 4-cycle, then it must be an iterated cone over the 4-cycle. This is nuclear. So focus on 4-cycle-free complexes. Then the -skeleton is either a complete graph, or two complete graphs glued along a clique. Complex induction argument based on the link of a vertex of C. Daniel Irving Bernstein Toric Varieties in Statistics 6 / 37

27 Next Steps - Unimodularity Question Given a simplicial complex C on [n] and an integer vector d = (d,..., d n ) with d i, is A C,d unimodular? Corollary (B.-Sullivant 05) If A C,d is unimodular, then C is nuclear. Question Let C and d be specified by the figure below. For which values of p and q is A C,d unimodular? 3 p q Daniel Irving Bernstein Toric Varieties in Statistics 7 / 37

28 Normality Let A N d n. We define: NA := {Ax : x N n } (Semigroup generated by columns of A) ZA := {Ax : x Z n } (Lattice generated by columns of A) R 0 A := {Ax : x R, x 0} (Cone generated by columns of A) Definition (Normality) We say that A is normal if NA = R 0 A ZA. If A is not normal and h R 0 A ZA \ NA the we say that h is a hole of NA. Daniel Irving Bernstein Toric Varieties in Statistics 8 / 37

29 Normality: Non-example The following matrix is not normal ( ) A = 0 3 because so ( ) is a hole. Note: ( ) = ( ) = ( ) R 0 A ZA. However, ( 0 3 ( ) 0 3 ( ) / NA. ) 0 0 / / Daniel Irving Bernstein Toric Varieties in Statistics 9 / 37

30 Normality Question When is A C,d normal? Observation If A C,d is normal, then so is A C := A C,(,...,). Terminology abuse C is normal means A C is normal Applications include: Integer table feasibility problem Toric fiber products for constructing Markov bases work best with normal A C,d (Rauh-Sullivant 04) Sequential importance sampling works best with normal A C,d We have some partial results towards classification of normal C Daniel Irving Bernstein Toric Varieties in Statistics 30 / 37

31 Known Classification Results - Normality Theorem (Sullivant 00) If C is a graph, then A C is normal if and only if C is free of K 4 -minors. Theorem (Bruns, Hemmecke, Hibi, Ichim, Ohsugi, Köppe, Söger 007-0) Let C be a complex whose facets are all m element subsets of [m]. Then A C,d is normal in precisely the following situations up to symmetry: At most two of the d v are greater than two m = 3 and d = (3, 3, a) for any a N 3 m = 3 and d = (3, 4, 4), (3, 4, 5) or (3, 5, 5). Theorem (Rauh-Sullivant 04) Let C be the four-cycle graph. Then A C,d is normal if d = (, a,, b) or d = (, a, 3, b) with a, b, N. Daniel Irving Bernstein Toric Varieties in Statistics 3 / 37

32 Corollary of Unimodular Classification Definition Let C be a simplicial complex on [n]. We say a facet of C that has n vertices is called a big facet. Proposition If C is a complex with a big facet, then C is normal if and only if unimodular. So our classification result on unimodular C immediately gives a classification of the normal C when C has a big facet. Daniel Irving Bernstein Toric Varieties in Statistics 3 / 37

33 Normality Preserving Operations Theorem (Sullivant 00) Normality of A C,d is preserved under the following operations on the simplicial complex Deleting vertices Contracting edges 3 Gluing two simplicial complexes along a common face 4 Adding or removing a cone or ghost vertex. Theorem (B.-Sullivant 05) Normality of A C,d is preserved when taking links of vertices of C. Daniel Irving Bernstein Toric Varieties in Statistics 33 / 37

34 Minimally Non-Normal Simplicial Complexes Question Which simplicial complexes are minimally non-normal with respect to the operations of deleting vertices, contracting edges, gluing two complexes along a facet, removing cone and ghost vertices, and taking links of vertices? Computational method: All simplicial complexes on 3 or fewer vertices are normal Choose two normal simplicial complexes C, D on n vertices. Create simplicial complex C on n vertices by attaching a new vertex v to C such that link v (C ) = D See if (non)normality of C can be certified by reducing to a smaller complex via our normality-preserving operations If not, check normality of C using Normaliz. If non-normal, then minimally non-normal Daniel Irving Bernstein Toric Varieties in Statistics 34 / 37

35 Minimally Non-Normal Simplicial Complexes We were able to use the computational method to determine normality on all complexes on up to 6 vertices So far, we know that the set of minimally non-normal simplicial complexes consists of: 0 sporadic complexes, obtained by computational method Two infinite families, obtained by theoretical means Daniel Irving Bernstein Toric Varieties in Statistics 35 / 37

36 Next Steps Develop new procedures for constructing normal C Develop methods for constructing holes of NA C Classify normal complexes within certain families (e.g., surfaces) Daniel Irving Bernstein Toric Varieties in Statistics 36 / 37

37 References Daniel Irving Bernstein and Seth Sullivant. Unimodular Binary Hierarchical Models. ArXiv:50.063, 05. Daniel Irving Bernstein and Seth Sullivant. Normal Binary Hierarchical Models ArXiv: , 05 Winfried Bruns, Raymond Hemmecke, Bogdan Ichim, Matthias Köppe and Christof Söger. Challenging Computations of Hilbert Bases of Cones Associated with Algebraic Statistics Exp. Math. 0, 5-33 (0) Mathias Drton, Bernd Sturmfels, Seth Sullivant. Lectures on Algebraic Statistics Birkhauser. Oberwolfach Seminars, Vol 39 Takayuki HIbi and Hidefumi Ohsugi Toric Ideals Arising from Contingency Tables Commutative Algebra and Combinatorics Ramanujan Mathematical Society Lecture Notes Series, Number 4, Ramanujan Math. Soc., Mysore, 007, pp Johannes Rauh and Seth Sullivant Lifting markov bases and higher codimension toric fiber products ArXiv: Bernd Sturmfels. Gröbner Bases and Convex Polytopes, volume 8 of University Lecture Series. American Mathematical Society, Providence, RI, 996. Seth Sullivant. Normal binary graph models. Ann. Inst. Statist. Math., 6(4):77 76, 00. Daniel Irving Bernstein Toric Varieties in Statistics 37 / 37