Existence of maximum likelihood estimators for 3 toric network models Sonja Petrovi´ c (University of Illinois at Chicago) Penn State University Toric geometry and applications Leuven, Belgium June 8, 2011 Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 1 / 17 / 23
Algebraic statistics Fact (Guiding principle) Many important statistical models correspond to algebraic or semi-algebraic sets of parameters. The geometry of these parameter spaces determines the behavior of widely used statistical inference procedures. A typical question: dimension, degree, singularities? Generators and Gr¨ obner bases: A typical question: the ideal of a given toric/secant/join variety? When a model is algebraic, use tools from algebraic geometry and computational algebra software packages ( 4ti2 , Macaulay2 , polymake , Singular , ...) Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 2 / 17 / 23
The Beta model for random undirected graphs In a pairwise comparison of n items, for every ordered pair ( i , j ): x i , j := the number of pairwise comparisons involving objects i and j in which i emerged as a winner. Fix N i , j := x i , j + x j , i , the total number of comparisons. Definition (Set of all possible outcomes) S n := { x i , j : i < j and x i , j ∈ { 0 , 1 , . . . , N i , j }} ⊂ N ( n 2 ) . Definition (Parametrization of the beta model) For each β ∈ R n : e β i + β j 1 p i , j = and p j , i = 1 − p i , j = ∀ i � = j . 1 + e β i + β j , 1 + e β i + β j The model M β for n vertices consists of all p i , j ’s of this form. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 3 / 17 / 23
The Beta model for random undirected graphs It is parametrized by the vertex-edge incidence matrix of a complete graph: 1 1 1 0 0 0 1 0 0 1 1 0 A 4 = 0 1 0 1 0 1 0 0 1 0 1 1 rows indexed by the vertices; columns indexed by ( i , j ) with i < j . Definition (The model polytope) S n := conv { A n x , x ∈ S n } Example Represent the graph with an edge { 1 , 2 } and a triple edge { 1 , 3 } as x = [1 , 3 , 0 , . . . , 0] T ∈ S n . Corresponding point in the model polytope is A n x = [4 , 1 , 3 , 0 , . . . ] . Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 4 / 17 / 23
MLE existence problem Given the graph x ∈ S n : Goal (Motivating problem: maximum likelihood estimation) Determine parameter values that “best explain” this data. This is done by maximizing the log-likelihood function: p x ij � MLE ( p ) := argmax p ∈M n ij . i < j Definition If MLE(p) is achieved on the boundary of the model M , it is called the extended MLE. In this case, the MLE is said not to exist. Problem Determine the points x for which MLE ( p ) does not exist. Nonexistence of the MLE implies that only certain entries of p are estimable. (i.e., certain linear combinations of the natural parameters) Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 5 / 17 / 23
A small example of data leading to a nonexistent MLE × 0 N 1 , 2 × e β i + β j (Recall M β : p i , j = 1+ e β i + β j . ) × N 3 , 4 0 × × 0 1 2 × 0 0.5 0.5 3 × 2 1 1 × 0.5 0.5 2 1 × 3 0.5 0.5 × 1 1 2 0 × 0.5 0.5 0 × Left : data exhibiting the above pattern, when N i , j = 3 for all i � = j . Right : table of the extended MLE of the estimated probabilities. Under natural parametrization, the supremum of the log-likelihood is achieved in the limit for any sequence of natural parameters { β ( k ) } of the form β ( k ) = ( − c k , − c k , c k , c k ), where c k → ∞ as k → ∞ . Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 6 / 17 / 23
Polyhedral methods Theorem (Standard statistical theory of exponential families) The MLE for an observed graph x exists if and only if A n x ∈ int( S n ) . A general algorithm for deciding this problem and finding the relevant facial sets are presented in Eriksson, Fienberg, Rinaldo, Sullivant (’06). Problem Understand the model polytope for M β . Extend to other random graph models. Rinaldo, Fienberg (May ’11) provide a more thorough treatment of algorithms for log-linear (toric) models. Rinaldo, P., Fienberg (May ’11) specialize these to network models. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 7 / 17 / 23
Motivation = size of random graph Example: The Collective Dynamics of Smoking in a Large Social Network (James Fowler) Node border= gender (red=female, blue=male). Arrow color = relation (purple=friend, green=spouse). Node color = smoking behavior (white=nonsmoker, gray=smoker); darker shades = more cigarettes consumed per day. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 8 / 17 / 23
The polytope of degree sequences Definition The polytope of degree sequences is P n := convhull ( { Ax , x ∈ G n } ) . Facet-defining inequalities of P n are known (Mehadev-Peled ’96). Theorem (Rinaldo-P.-Fienberg) Let x ∈ S n be the observed vector of edge counts. The MLE exists if and only if x j , i x i , j � � + ∈ int ( P n ) , i = 1 , . . . , n . N i , j N i , j j < i j > i Example (Stanley) f ( P 8 ) = (334982 , 1726648 , 3529344 , 3679872 , 2074660 , 610288 , 81144 , 3322 , 1) . We used polymake for the computations on small polytopes. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 9 / 17 / 23
Facial sets of the model polytope Proposition (Rinaldo-P.-Fienberg) A point y belongs to the interior of some face F of P n if and only if there exists a set F ⊂ { ( i , j ) , i < j } such that for any p = { p i , j : i < j , p i , j ∈ [0 , 1] } satisfying y = A n p , p i , j ∈ { 0 , 1 } if ( i , j ) �∈ F and p i , j ∈ (0 , 1) if ( i , j ) ∈ F . F is called a facial set of S n , and F c a co-facial set. The MLE does not exist for the graph x if and only if the set { ( i , j ): i < j , x i , j = 0 or N i , j } contains a co-facial set. Facial sets specify which probability parameters are estimable: only the probabilities { p i , j , ( i , j ) ∈ F} . Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 10 / 17 / 23
Example: Co-facial sets for nonexistent MLEs × 0 × 0 N 1 , 2 × N 1 , 2 × 0 0 × N 3 , 4 N 3 , 2 × 0 × N 4 , 2 × × 0 0 0 × 0 0 N 1 , 2 × N 1 , 2 × 0 N 1 , 3 × N 1 , 3 N 2 , 3 × N 4 , 1 × × × N 1 , 2 0 × 0 0 N 2 , 3 × N 2 , 4 × Table: Co-facial sets for P 4 (empty cells indicate any entry values). Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 11 / 17 / 23
Two related toric models where main Theorem applies Definition (Random graphs with fixed degree sequence) In the special case when N ij = 1 , the support S n reduces to G n := { 0 , 1 } ( n 2 ) , undirected simple graphs on n nodes. Corollary (RPF) A conjecture in Chatterjee-Diaconis-Sly (’10) is true: for the random graph model, the MLE exists if and only if d ( x ) ∈ int P n . Definition (The Rasch model) A random bipartite graph model, the support being G k , l , the set of bipartite graphs on k and l vertices. Theorem (RPF) The MLE of the Rasch model parameters exists if and only if d ( x ) ∈ int P p , q , the polytope of bipartite degree sequences. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 12 / 17 / 23
Extensions (1) Removing the sampling constraint: Let quantities N i , j be random! Theorem (Thanks to Haase and Yu) The model polytope has 3 n facets, and is obtained from the product of simplices by removing the vertices { e i × e ′ i } , i = 1 , . . . , n. (2) Specialize (1) to directed graphs without multiple edges. This is the Bradley-Terry model for pairwise comparisons. Theorem (Zermelo ’29, Ford ’57) If the graph is strongly connected, then the MLE exists. Algorithms for detecting co-facial sets still apply. The matrix of the � n � model polytope has dimension ( + n ) × n ( n − 1). 2 Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 13 / 17 / 23
Extensions (3) A directed random graph model used in social networking: the p 1 model (Holland-Leinhardt ’81). � n � The model polytope is the Minkowski sum of polytopes. 2 Example (n=4) 4 10 = 1 , 048 , 576 different graphs x. Three cases of the p 1 model: 1 There are 225 , 025 points A 4 x, and the MLE exists for 7 , 983 . 2 349 , 500 , the MLE exists in 12 , 684 cases 3 583 , 346 , the MLE never exists. Theorem (Rinaldo-P.-Fienberg) Sufficient conditions for MLE existence, with large probability, as n grows. In the case of fixed degree sequence graphs, our asymptotic results improve those of Chatterjee-Diaconis ’11. Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/ ∼ petrovic June 8, 2011 14 / 17 / 23
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