Existence of maximum likelihood estimators for 3 toric network - - PowerPoint PPT Presentation

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¨

• bner 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): xi,j := the number of pairwise comparisons involving objects i and j in which i emerged as a winner. Fix Ni,j := xi,j + xj,i, the total number of comparisons. Definition (Set of all possible outcomes) Sn := {xi,j : i < j and xi,j ∈ {0, 1, . . . , Ni,j}} ⊂ N(n

2).

Definition (Parametrization of the beta model) For each β ∈ Rn: pi,j = eβi+βj 1 + eβi+βj and pj,i = 1 − pi,j = 1 1 + eβi+βj , ∀i = j. The model Mβ for n vertices consists of all pi,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:

A4 =     1 1 1 1 1 1 1 1 1 1 1 1    

rows indexed by the vertices; columns indexed by (i, j) with i < j. Definition (The model polytope) Sn := conv {Anx, x ∈ Sn} Example Represent the graph with an edge {1, 2} and a triple edge {1, 3} as x = [1, 3, 0, . . . , 0]T ∈ Sn. Corresponding point in the model polytope is Anx = [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 ∈ Sn: Goal (Motivating problem: maximum likelihood estimation) Determine parameter values that “best explain” this data. This is done by maximizing the log-likelihood function: MLE(p) := argmaxp∈Mn

• i<j

pxij

ij .

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

× N1,2 × × N3,4 × (Recall Mβ: pi,j =

eβi +βj 1+eβi +βj .)

× 1 2 3 × 2 1 2 1 × 3 1 2 × × 0.5 0.5 1 × 0.5 0.5 0.5 0.5 × 1 0.5 0.5 × Left: data exhibiting the above pattern, when Ni,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) = (−ck, −ck, ck, ck), where ck → ∞ 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 Anx ∈ int(Sn). 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 Mβ. 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 Pn := convhull ({Ax, x ∈ Gn}) . Facet-defining inequalities of Pn are known (Mehadev-Peled ’96). Theorem (Rinaldo-P.-Fienberg) Let x ∈ Sn be the observed vector of edge counts. The MLE exists if and

• nly if
• j<i

xj,i Ni,j +

• j>i

xi,j Ni,j ∈ int(Pn), i = 1, . . . , n.

Example (Stanley) f (P8) = (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 Pn if and only if there exists a set F ⊂ {(i, j), i < j} such that for any p = {pi,j : i < j, pi,j ∈ [0, 1]} satisfying y = Anp, pi,j ∈ {0, 1} if (i, j) ∈ F and pi,j ∈ (0, 1) if (i, j) ∈ F. F is called a facial set of Sn, and Fc a co-facial set. The MLE does not exist for the graph x if and only if the set {(i, j): i < j, xi,j = 0 or Ni,j} contains a co-facial set. Facial sets specify which probability parameters are estimable:

• nly the probabilities {pi,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

× N1,2 × × N3,4 × × N1,2 × N3,2 × N4,2 × × N1,2 × N1,3 × N4,1 × × N1,2 × N1,3 N2,3 × × × N1,2 × N2,3 × N2,4 × Table: Co-facial sets for P4 (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 Nij = 1, the support Sn reduces to Gn := {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 Pn.

Definition (The Rasch model) A random bipartite graph model, the support being Gk,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 Pp,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 Ni,j be random! Theorem (Thanks to Haase and Yu) The model polytope has 3n facets, and is obtained from the product of simplices by removing the vertices {ei × 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 model polytope has dimension ( n

2

• + n) × n(n − 1).

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 p1 model (Holland-Leinhardt ’81). The model polytope is the Minkowski sum of n

2

• polytopes.

Example (n=4) 410 = 1, 048, 576 different graphs x. Three cases of the p1 model:

1 There are 225, 025 points A4x, 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

Defining ideals of the models are known

Theorem (Sturmfels-Welker, Theorem 7.7, arXiv:1101.1597, Jan ’11.) The Bradley-Terry model is toric. It is defined by the unimodular Lawrence ideal whose generators are circuits of a bidirected graph. Theorem (Ogawa-Hara-Takemura, Proposition1, arXiv:1102.2583, Feb’11.) Markov basis of the Beta model is given by the Graver basis of the toric ideal of an undirected graph.

In addition, they provide an algorithm for generating moves on the fly.

Theorem (P.-Rinaldo-Fienberg) The toric ideal of the p1 random graph model on n nodes is the multi-homogenous piece of the ideal generated mainly by the defining equations for the edge subring of a bipartite graph.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 15 / 17 / 23

The story of 80,000 Markov moves

Using the natural toric parametrization of the p1 model, 4ti2 computed a minimal generating set for the 4-node graph: 80, 610 binomials. Using our parametrization, we obtain 77 minimal generators. Using our Theorem and respecting the multi-homogeneous constraint, we were able to explain all the generators in terms of the essential 10.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 16 / 17 / 23

Algebraic statistics – general references

• Diaconis, Sturmfels ’98: Gr¨
• bner bases for exact conditional tests.
• Pistone, Wynn ’96: Use Gr¨
• bner bases to study confounding in design
• f experiments.
• Pistone, Riccomagno, Wynn ’01, Algebraic statistics: computational

commutative algebra in statistics.

• Gibilisco, Riccomagno, Rogantin, Wynn ’09: Algebraic and Geometric

methods in statistics.

• Drton, Sturmfels, Sullivant ’09: Lectures in algebraic statistics.
• Viana, Wynn ’10: Algebraic Methods in Statistics and Probability, II
• Applications to computational biology (Pachter, Sturmfels ’05)
• P., Slavkovic, Algebraic statistics, in: International Encyclopedia of

Statistical Science, ed. M. Lovric, Springer 2011.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 17 / 17 / 23

The p1 random graph model

The p1 random graph model (Holland-Leinhardt)

n nodes, random occurrence of directed edges. Each pair {i, j} modeled independently:

pij(0, 0) = no edge pij(1, 0) = edge from i to j pij(0, 1) = edge from j to i pij(1, 1) = bidirected edge between i and j.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 18 / 17 / 23

The p1 random graph model

The p1 random graph model (Holland-Leinhardt)

n nodes, random occurrence of directed edges. Each pair {i, j} modeled independently:

pij(0, 0) = no edge pij(1, 0) = edge from i to j pij(0, 1) = edge from j to i pij(1, 1) = bidirected edge between i and j.

pij := (pij(0, 0), pij(1, 0), pij(0, 1), pij(1, 1)) ∈ ∆3 ⊂ R4.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 18 / 17 / 23

The p1 random graph model

The p1 random graph model (Holland-Leinhardt)

n nodes, random occurrence of directed edges. Each pair {i, j} modeled independently:

pij(0, 0) = no edge pij(1, 0) = edge from i to j pij(0, 1) = edge from j to i pij(1, 1) = bidirected edge between i and j.

pij := (pij(0, 0), pij(1, 0), pij(0, 1), pij(1, 1)) ∈ ∆3 ⊂ R4. Definition The model Mn is the image of the simplex under the polynomial map ϕn : C[pij(∗, ∗)] → C[λij, αi, βi, θ, ρij] pij(1, 0) → λijαiβjθ, pij(0, 0) → λij, pij(0, 1) → λijαjβiθ, pij(1, 1) → λijαiαjβiβjθ2ρij.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 18 / 17 / 23

The p1 random graph model

Random walk on a fiber

The problem: determine whether the model fits this data. Definition A fiber of a graph G is F(G) := {H such that ϕn(G) = ϕn(H)}. Standard asymptotics not applicable = ⇒ random walk on a fiber. Alternative, non-asymptotic approach to testing goodness of fit. Definition A Markov basis is a set of moves that is guaranteed to connect each fiber.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 19 / 17 / 23

The p1 random graph model

The Fundamental Theorem of Markov Bases

Definition The toric ideal of the model is the kernel of the parametrization map ϕn. Example ker ϕ3 = (p12(1, 0)p23(1, 0)p13(0, 1) − p12(0, 1)p23(0, 1)p13(1, 0)) Theorem (Diaconis and Sturmfels ’98) A set of moves is a Markov basis if and only if the corresponding binomials generate the toric ideal of the model. Computed in practice using the software 4ti2 (Hemmecke et. al.) Problem Describe a Markov basis for n-node network for any n.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 20 / 17 / 23

The p1 random graph model

A classical construction: Consider a bipartite graph with vertices α1, . . . , αn and β1, . . . , βn. Edge subring of the graph is the monomial subalgebra generated by the edges: pij(1, 0) = αiβj, pij(0, 1) = αjβi (Isomorphic to the special fiber ring of the edge ideal of the graph) The toric ideal of the edge subring is the kernel of this map. Theorem (Ohsugi-Hibi, Villarreal, . . . ) All polynomial relations among the edges pij are generated by the cycles in the bipartite graph. p14(1, 0)p23(1, 0)p24(0, 1)− p12(1, 0)p24(1, 0)p34(0, 1) = 0

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 21 / 17 / 23

The p1 random graph model

Definition A simplified model is obtained by ignoring λij. Theorem (P.-Rinaldo-Fienberg) The toric ideal of the simplified model is generated by binomials of the form pij(1, 0)pij(0, 1) − pij(1, 1) together with the defining equations for the edge subring of a bipartite graph. p14(1, 0)p23(1, 0)p24(0, 1)− p12(1, 0)p24(1, 0)p34(0, 1) = 0 The corresponding Markov move for the simplified models:

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 22 / 17 / 23

The p1 random graph model

Markov bases for p1 models

Theorem (P.-Rinaldo-Fienberg) The toric ideal of the p1 random graph model on n nodes is the multi-homogenous piece (with respect to each pair i, j) of the ideal generated mainly by the defining equations for the edge subring of a bipartite graph.

p14(1, 0)p23(1, 0)p24(0, 1) − p12(1, 0) p24(1, 0)p34(0, 1) = 0

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 23 / 17 / 23

The p1 random graph model

Markov bases for p1 models

Theorem (P.-Rinaldo-Fienberg) The toric ideal of the p1 random graph model on n nodes is the multi-homogenous piece (with respect to each pair i, j) of the ideal generated mainly by the defining equations for the edge subring of a bipartite graph.

p12(0, 0)p14(1, 0)p23(1, 0)p24(0, 1)p34(0, 0)− p12(1, 0)p14(0, 0)p23(0, 0)p24(1, 0)p34(0, 1) = 0

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 23 / 17 / 23

The p1 random graph model

Markov bases for p1 models

Theorem (P.-Rinaldo-Fienberg) The toric ideal of the p1 random graph model on n nodes is the multi-homogenous piece (with respect to each pair i, j) of the ideal generated mainly by the defining equations for the edge subring of a bipartite graph.

p12(0, 0)p14(1, 0)p23(1, 0)p24(0, 1)p34(0, 0)− p12(1, 0)p14(0, 0)p23(0, 0)p24(1, 0)p34(0, 1) = 0

Homogenizing generators of the simplified model Moves incorporate sampling constraints: applicable to network data Efficiency improvements over existing methods for generating moves.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 23 / 17 / 23

Existence of MLE for p1 - details

Polyhedral conditions

In the case ρij = ρ + ρi + ρj, the matrix representing ϕn is A = λ12 λ13 λ23 θ α1 α2 α3 β1 β2 β3 ρ ρ1 ρ2 ρ3 p1,2 p1,3 p2,3 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 2 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

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Existence of MLE for p1 - details

Polyhedral conditions

Definition The marginal polytope is PA = conv({t : t = Ax, x ∈ Xn}). PA =

• i<j

conv(Ai,j), where Ai,j is the submatrix corresponding to pij. PA is the convex hull of the set of all possible observable marginals From standard statistical theory of exponential families, the MLE ˆ p exists if and only if the vector of margins t = Ax belongs to ri(PA).

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 25 / 17 / 23

Existence of MLE for p1 - details

Two problems:

1

decide whether t belongs to ri(PA)

2

compute supp(ˆ p), supp(x) = {i : xi = 0}. (the extended MLE)

Combinatorial complexity of Minkowki sums! The supports of the points on the boundary of PA are determined by facial sets of the marginal cone CA = cone(A): Theorem (Rinaldo-P.-Fienberg) Let t ∈ PA. Then, t ∈ ri(PA) if and only if t ∈ ri(CA). Simpler face lattice Algorithms for deciding whether t ∈ ri(CA) and for finding the facial set corresponding to the face F of CA such that t ∈ ri(F) are presented in Eriksson, Fienberg, Rinaldo, Sullivant (’06)

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Existence of MLE for p1 - details

n ρi,j = 0 ρi,j = ρ ρ = ρi + ρj 2n(n − 1) 3 62 62 62 12 4 1,862 2,415 3,086 24 5 88,232 158,072 347,032 40 Table: Number of vertices for the polytopes PA for different specifications of the p1 model and different network sizes. Computations carried out using minksum. The last column = the number of columns of the design matrix A, which correspond to the number of generators of CA.

n ρi,j = 0 ρi,j = ρ ρ = ρi + ρj Facets Dim. Ambient Dim. Facets Dim. Ambient Dim. Facets Dim. Ambient Dim. 3 30 7 9 56 8 10 15 10 13 4 132 12 14 348 13 15 148 16 19 5 660 18 20 3032 19 21 1775 23 26 6 3181 25 27 94337 26 28 57527 31 34

Table: Number of facets, dimensions and ambient dimensions of the the cones CA for different specifications of the p1 model and different network sizes.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 27 / 17 / 23

Existence of MLE for p1 - details

How often does the MLE exist?

Networks as n × n incidence matrices:

2 4 × p12(1, 0) p13(1, 0) p12(0, 1) × p23(1, 0) p13(0, 1) p23(0, 1) × 3 5 Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 28 / 17 / 23

Existence of MLE for p1 - details

How often does the MLE exist?

Networks as n × n incidence matrices:

2 4 × p12(1, 0) p13(1, 0) p12(0, 1) × p23(1, 0) p13(0, 1) p23(0, 1) × 3 5

Patterns of 0’s leading to nonexistent MLE:

Maximal observable number of 1’s: a row or column sum is n − 1 If row i sums to n − 1, then ˆ pij(1, 0) = 1 for all j, which implies that ˆ pij(0, 1) = ˆ pij(0, 0) = 0, for all j.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 28 / 17 / 23

Existence of MLE for p1 - details

How often does the MLE exist?

Networks as n × n incidence matrices:

2 4 × p12(1, 0) p13(1, 0) p12(0, 1) × p23(1, 0) p13(0, 1) p23(0, 1) × 3 5

Patterns of 0’s leading to nonexistent MLE:

Maximal observable number of 1’s: a row or column sum is n − 1 If row i sums to n − 1, then ˆ pij(1, 0) = 1 for all j, which implies that ˆ pij(0, 1) = ˆ pij(0, 0) = 0, for all j. Minimal observable number of 1’s: a row or column sum is 0 If row i has a zero sum, then ˆ pij(1, 0) = ˆ pij(1, 1) = 0, for all j.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 28 / 17 / 23

Existence of MLE for p1 - details

How often does the MLE exist?

Networks as n × n incidence matrices:

2 4 × p12(1, 0) p13(1, 0) p12(0, 1) × p23(1, 0) p13(0, 1) p23(0, 1) × 3 5

Patterns of 0’s leading to nonexistent MLE:

Maximal observable number of 1’s: a row or column sum is n − 1 If row i sums to n − 1, then ˆ pij(1, 0) = 1 for all j, which implies that ˆ pij(0, 1) = ˆ pij(0, 0) = 0, for all j. Minimal observable number of 1’s: a row or column sum is 0 If row i has a zero sum, then ˆ pij(1, 0) = ˆ pij(1, 1) = 0, for all j. General rule for last case unknown. 4 patterns for n = 4, e.g.:

    × × × ×     ,     × × × ×     ,     × × × ×     ,

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Existence of MLE for p1 - details

Facet description for nonexistent MLE

Let B be the submatrix of the design matrix A using only pij(1, 0) and pij(0, 1) columns. When n = 3:

B = 2 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 1 1 3 7 7 7 7 7 5

Sufficient to look at facial sets of pointed polyhedral cone cone(B) (columns of B are affinely independent)

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 29 / 17 / 23

Existence of MLE for p1 - details

Facet description for nonexistent MLE

Let B be the submatrix of the design matrix A using only pij(1, 0) and pij(0, 1) columns. When n = 3:

B = 2 6 6 6 6 6 4 1 1 1 1 1 1 1 1 1 1 1 1 3 7 7 7 7 7 5

Sufficient to look at facial sets of pointed polyhedral cone cone(B) (columns of B are affinely independent) Based on computations carried out in polymake for networks of size up to n = 10: Theorem (Rinaldo, P., Yu, Haase) For a network on n nodes, cone(B) has 3n facets, 2n of which correspond to patterns of zeros leading to a zero row or column margin, and the remaining n to patterns of zeros which cause the MLE not to exist without inducing zero margins.

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 29 / 17 / 23

Existence of MLE for p1 - details

THE END

Sonja Petrovi´ c (UIC – Penn State) www.math.uic.edu/∼petrovic June 8, 2011 30 / 17 / 23