
Laura Matusevich (Texas A&M) IMA postdoc seminar: Combinatorics of binomial primary decomposition 
Abstract: Using examples, I will illustrate the main elements needed to
explicitly describe the primary components of a binomial ideal,
emphasizing the connections to combinatorics and (hypergeometric)
differential equations. This is joint work with Alicia Dickenstein and
Ezra Miller. 
Elizabeth S. Allman (University of Alaska) 
Phylogenetic models: algebra and evolution 
Abstract: Molecular phylogenetics is concerned with inferring evolutionary
relationships (phylogenetic trees) from biological sequences (such as
aligned DNA sequences for a gene shared by a collection of species).
The probabilistic models of sequence evolution that underly statistical
approaches in this field exhibit a rich algebraic structure.
After an introduction to the inference problem and phylogenetic
models, this talk will survey some of the highlights of current
algebraic understanding. Results on the important statistical issue
of identifiability of phylogenetic models will be emphasized, as the
algebraic viewpoint has been crucial to obtaining such results. 
Niko Beerenwinkel (Harvard University) 
Gene interactions and the geometry of fitness landscapes 
Abstract: The relationship between the shape of a fitness landscape and the underlying gene interactions, or epistasis, has been extensively studied in the twolocus case. Epistasis has been linked to biological important properties such as the advantage of sex. Gene interactions among multiple loci are usually reduced to twoway interactions. Here, we present a geometric theory of shapes of fitness landscapes for multiple loci. We investigate the dynamics of evolving populations on fitness landscapes and the predictive power of the geometric shape for the speed of adaptation. Finally, we discuss applications to fitness data from viruses and bacteria.

Joe Buck (Lockheed Martin Coherent Technologies ) 
IMA/MCIM Industrial problems seminar: Optical synthetic aperture imaging 
Abstract: The spatial resolution of a conventional imaging ladar system is
constrained by the diffraction limit of the telescope's aperture. At
Lockheed Martin Coherent Technologies (LMCT) we are implementing
techniques known as syntheticaperture imaging laser radar (SAIL), which
employs aperture synthesis with coherent laser radar to overcome the
diffraction limit and achieve fineresolution, longrange,
twodimensional imaging with modest aperture diameters. I will discuss
the results of my experiments while at The Aerospace Corporation which
represent the first optical synthetic aperture images of a fixed,
diffusely scattering target with moving aperture, as well as the current
research program being developed at LMCT. 
Marta Casanellas (Polytechnical University of Cataluña (Barcelona)) 
Using algebraic geometry for phylogenetic reconstruction 
Abstract: Many statistical models of evolution can be viewed as
algebraic varieties. The generators of the ideal associated to a model
and a phylogenetic tree are called invariants. The invariants of an
statistical model of evolution should allow to determine what is the
tree formed by a set of living species.
We will present a method of phylogenetic inference based on invariants
and we will discuss why algebraic geometry should be considered as a
powerful tool for phylogenetic reconstruction. The performance of the
method has been studied for quartet trees and the Kimura 3parameter
model and it will be compared to widely known phylogenetic
reconstruction methods such as Maximum likelihood estimate and
NeighborJoining. 
Gheorghe Craciun (University of Wisconsin) 
Stability and instability in polynomial equations
arising from complex chemical reaction networks: some underlying
mathematics 
Abstract: Chemical reaction network models give rise to
polynomial
dynamical systems that are usually high dimensional,
nonlinear, and
have many unknown parameters. Due to the presence of these
unknown
parameters (such as reaction rate constants) direct numerical
simulation of the chemical dynamics is practically
impossible. On
the other hand, we will show that important properties of
these
systems are determined only by the network structure, and do
not
depend on the unknown parameters. Also, we will show how some
of
these results can be generalized to systems of polynomial
equations
that are not necessarily derived from chemical kinetics. In
particular, we will point out connections with classical
problems
in algebraic geometry, such as the real Jacobian conjecture.
This
talk describes joint work with Martin Feinberg, and can be
regarded
as a continuation of his earlier talk. 
Kenneth R. Driessel (Iowa State University) 
Real algebraic geometry tutorial: Real rings (continued) 
Abstract: Recall that we
started to talk about real rings on March 1. The topic
was new for us at that time.
Also recall that a commutative ring with identity is "real"
if the only representation of zero as the sum of squares
is the trivial one.
Since we did not meet during the last two weeks,
I shall include an substantial review as part of
our discussion.
I shall mainly follow the material in the chapter
"Real Rings" in the book "Positive Polynomials"
by Prestel and Delzell.
Our objective will be a proof of an absrtact version of
the real Nullstellensatz. 
Mathias Drton (University of Chicago) 
Multiple solutions to the likelihood equations in
the BehrensFisher problem 
Abstract: The BehrensFisher problem concerns testing the statistical hypothesis
of equality of the means of two normal populations with possibly
different variances. This problem
furnishes one of the simplest statistical models for which the likelihood
equations may have more than one real solution. In fact, with
probability one, the equations have either one or three real solutions.
Using the cubic discriminant, we study the largesample probability of
one versus three solutions. 
Nicholas Eriksson (Stanford University) 
Metric learning for phylogenetic invariants 
Abstract: We introduce new methods for phylogenetic tree construction by using
machine learning to optimize the power of phylogenetic invariants.
Phylogenetic invariants are polynomials in the joint probabilities
which vanish under a model of evolution on a phylogenetic tree. We
give algorithms for selecting a good set of invariants and for
learning a metric on this set of invariants which optimally
distinguishes the different models. Our learning algorithms involve
semidefinite programming on data simulated over a wide range of
parameters. Simulations on trees with four leaves under the
JukesCantor and Kimura 3parameter models show that our method
improves on other uses of invariants and is competitive with
neighborjoining. Our main biological result is that the trained
invariants can perform substantially better than neighbor joining on
quartet trees with short interior edges.
This is joint work with Yuan Yao (Stanford). 
Martin Feinberg (Ohio State University) 
Stability and instability in polynomial equations
arising from complex chemical reaction networks: the big picture 
Abstract: In nature there are millions of distinct networks of chemical reactions that might present themselves for study at one time or another. Written at the level of elementary reactions taken with classical mass action kinetics, each new network gives rise to its own (usually large) system of polynomial equations for the species concentrations. In this way, chemistry presents a huge and bewildering array of polynomial systems, each determined in a precise way by the underlying network up to parameter values (e.g., rate constants). Polynomial systems in general, even simple ones, are known to be rich sources of interesting and sometimes wild dynamical behavior. It would appear, then, that chemistry too should be a rich source of dynamical exotica.
Yet there is a remarkable amount of stability in chemistry. Indeed, chemists and chemical engineers generally expect homogeneous isothermal reactors, even complex ones, to admit precisely one (globally attractive) equilibrium. Although this tacit doctrine is supported by a long observational record, there are certainly instances of homogeneous isothermal reactors that give rise, for example, to multiple equilibria. The vast landscape of chemical reaction networks, then, appears to have wide regions of intrinsic stability (regardless of parameter values) punctuated by far smaller regions in which instability might be extant (for at least certain parameter values).
In this talk, I will present some recent joint work with Gheorghe Craciun that goes a long way toward explaining this landscape — in particular, toward explaining how biological chemistry "escapes" the stability doctrine to (literally) "make life interesting." A subsequent talk by Craciun will emphasize more mathematical detail. 
Stephen E. Fienberg (CarnegieMellon University) 
Algebraic statistics and the analysis of contingency tables: Old wine in new bottles? 
Abstract: The past decade has seen considerable interest in the reformulation of statistical models and methods for the analysis of contingency tables using the language and results of algebraic and polyhedral geometry. But as algebraic statistics has developed, new ideas have emerged that have changed how we view a number of statistical problems. This talk reviews some of these recent advances and suggests some challenges for collaborative research, especially those involving large scale databases.

Robert M. Fossum (University of Illinois at UrbanaChampaign) 
Subspace arrangements in theory and practice 
Abstract: A subspace arrangement is a union of a finite number of subspaces of a
vector space. We will discuss the importance of subspace arrangements first
as mathematical objects and now as a popular class of models for
engineering.
We will then introduce some of new theoretical results that were motivated
from practice. Using these results we will address the computational issue
about how to extract subspace arrangements from noisy or corrupted data.
Finally we will turn to the importance of subspace arrangements by briefly
discussing the connections to sparse representations, manifold learning,
etc... 
Martin Golubitsky (University of Houston) 
Math Matters  IMA Public Lecture: Patterns Patterns Everywhere

Abstract: Regular patterns appear all around us: from vast geological formations to the ripples in a vibrating coffee cup, from the gaits of trotting horses to tongues of flames, and even in visual hallucinations. The mathematical notion of symmetry is a key to understanding how and why these patterns form. In this lecture Professor Golubitsky will show some of these fascinating patterns and explain how mathematical symmetry enters the picture. 
Mordechai Katzman (University of Sheffield) 
Algebraic geometry and applications seminar: Counting monomials 
Abstract: The contents of this elementary talk grew out of my need to explain to
nonmathematicians what I do for a living.
I will pose (and solve) two old chessboard enumeration problems and a new
problem. We will solve these by counting certain monomials, and this will
naturally lead us to the notion of Hilbert functions. With these examples in
mind, we will try and understand the simplest of monomial ideals, namely, edge
ideals, and discover that these are not simple at all! On the way we will
discover a new numerical invariant of forests. 
Markus Kirkilionis (University of Warwick) 
Modelling with massaction kinetics and beyond 
Abstract: Massaction kinetics is a powerful tool to describe events created by collission of molecules or individuals in a wellmixed environment giving them locally the same probability to meet each other. Moreover this probability is only dependent on the concentration of the mutual partners.
Mass action systems can be found in chemistry, cell biology, but also game theory and economics. Mathematically this gives rise to dynamical systems of a special type, more specific of polynomial type. I will give an overview how this property can be used to determine different types of bifurcations, for example the ocurrence of bistability, or oscillations via a Hopf bifurcation. All tools will be borrowing methods from algebraic geometry. Finally I will give an outlook what usually goes wrong in the modelling part while using massaction kinetics if biochemical or cellular molecular events are considered. Finally the talk ends with a fresh look on massaction kinetics applied to a spatial setting. 
Reinhard Laubenbacher (Virginia Polytechnic Institute and State University) 
Polynomial dynamical systems over finite fields, with
applications to modeling and simulation of biological networks 
Abstract: Timediscrete dynamical systems with a finite state space
have been used as models of biological systems since the
use/invention of cellular automata by von Neumann in his attempt to
model a selfreplicating organism in the 1950s. More recently, they
have appeared as models of a variety of biological systems, from
gene regulatory networks to largescale epidemiological networks. This
talk will focus on theoretical and computational results about
polynomial dynamical systems using tools from computational
algebra and algebraic geometry. 
Gennady Lyubeznik (University of Minnesota Twin Cities) 
IMA postdoc seminar: The minimum number of settheoretic defining equations of
algebraic varieties 
Abstract: Given an algebraic variety in affine or projective space, what
is the minimum number of equations that define this variety
settheoretically? This is a very difficult problem in general; no
algorithm to compute this minimum number is currently known. We will
discuss some techniques for upper and lower bounds and consider some
interesting specific examples. 
František Matúš (Institute of Information Theory and Automation) 
Generalized maximum likelihood estimates for exponential families 
Abstract: Exponential families underpin numerous models of statistics
and information geometry that have significant applications.
For a standard full exponential family, or its canonically convex
subfamily, if the corresponding likelihood function from a sample
has a maximizer t* then, by the maximum likelihood principle, the
data are judged to be generated by the probability measure P* from
the family that is parameterized by t*. Since the likelihood depends
on data only through their mean, in this way the mean is mapped to
P*. In a joint work with Imre Csiszar, Budapest, we study an
extension of this mapping, the generalized maximum likelihood
estimator. It assigns to each point of the space at which the
likelihood function is bounded above, a probability measure from
the closure of the family in variation distance. A detailed
description, complete characterization of domain and range, and
additional results will be presented, not imposing any regularity
assumptions. 
Jason Morton (University of California) 
Geometry of rank tests 
Abstract: We investigate the polyhedral geometry of conditional probability and
undirected graphical models, developing new statistical procedures
called convex rank tests. The polytope associated to an undirected
graphical conditional independence model is the graph associahedron.
The convex rank test defined by the dual semigraphoid to the ncycle
graphical model is applied to microarray data analysis to detect
periodic gene expression.

Sarah Olson (North Carolina State University) 
Supervised learning artifical neural network algorithms
for optimizing mechanical properties of elastinlike polypeptide
hydrogels for cartilage repair 
Abstract: Joint work with Dana L. Nettles^{3}, Kimberly Trabbic Carlson^{3},
Ashutosh Chilkoti^{3}, Lori A. Setton^{3,4}, Mansoor A. Haider^{1,2}
Elastinlike polypeptide (ELP) hydrogels are a class of biomaterials that
have potential utility as a biocompatible scaffold for filling defects due
to osteoarthritis and for regenerating cartilage. Because of the facility
to genetically engineer elastin sequence, there are almost endless
possible configurations of ELPs and conformations of the networks formed
after crosslinking. ELP biomaterial function will exhibit a complex
dependence on these polymer characteristics that impacts properties
expected to affect cartilage regeneration, such as mechanical load
support. These complex structurefunction relationships for crosslinked
ELP hydrogels are not well described. A method for predicting the
mechanical properties of ELP hydrogels was developed based on structural
properties and Supervised Artificial Neural Network (ANN) modeling. The
ANN Model used concentration, molecular weight, crosslink density, and
sample number to predict the dynamic shear modulus and loss angle of the
hydrogels. The ANN was implemented in a custom compiled code based on the
Scaled Conjugate Gradient minimization algorithm and a Monte Carlo Method
was used to expand the dataset. The ANN was trained using a varying
subsets of the full dataset (22 formulations), with the complementary
subset used for validation. Trained networks demonstrated excellent
accuracy in prediction of hydrogel dynamic shear modulus at physiological
temperature, based on polymer design and predictions were robust with
respect to statistical variations. The results are used to show the
validity of an intermediate screening process using ANNs to obtain the
optimal mechanical properties for the ELP.
^{1} Biomathematics Graduate Program, North Carolina State University, ^{2}
Department of Mathematics, North Carolina State University, ^{3} Department
of Biomedical Engineering, Duke University, ^{4} Department of Surgery, Duke
Medical Center

Lior Pachter (University of California) 
From Drosophila and transposable elements to phylogenetic networks and associahedra 
Abstract: We begin with an overview of the Drosophila genome project, whose goal is the sequencing and comparison of 12 fruit fly genomes. In particular, we describe some of the dynamic behavior of transposable elements. These are selfreplicating sequences that play a major role in shaping the structure and function of genomes. Our methods for studying transposable elements lead naturally to the analysis of split systems and their associated phylogenetic networks. We explain why the tessellation of \$\overline{M}_0^n\$ by associahedra is a natural candidate for the space of phylogenetic networks, and explain the relevance of this observation to the analysis of the popular neighbornet algorithm used for studying split systems. We discuss various aspects of the neighbornet algorithm, including its interpretation as a greedy algorithm for the travelingsalesman problem, how to obtain statistically meaningful parameters, and how to interpret its output. The application of neighbornet to the split system we obtain from transposable elements in Drosophila reveals interesting insights about a set of species that may have undergone lineage sorting. This is joint work with Anat Caspi and Dan Levy. 
Sonja Petrovic (University of Kentucky) 
Toric ideals of phylogenetic invariants for the general groupbased model on claw trees 
Abstract: We address the problem of studying the toric ideals of
phylogenetic invariants for a general groupbased model on an
arbitrary claw tree. We focus on the group _{2} and
choose a natural recursive approach that extends to other
groups. The study of the lattice associated with each
phylogenetic ideal produces a list of circuits that generate
the corresponding lattice basis ideal. In addition, we
describe explicitly a quadratic lexicographic Gröbner basis
of the toric ideal of invariants for the claw tree on an
arbitrary number of leaves. Combined with a result of Sturmfels
and Sullivant, this implies that the phylogenetic ideal of
every tree for the group _{2} has a quadratic
Gröbner basis. Hence, the coordinate ring of the toric
variety is a Koszul algebra.
This is joint work with Julia Chifman, University of Kentucky. 
Giovanni Pistone (Politecnico di Torino) 
Information geometry and algebraic statistics 
Abstract: Recent presentations of Information Geometry (IG), e.g. Amari and Nagaoka (2000), consider general statistical models and general sample spaces. However, the seminal discussion by Cenkov (transl. 1982) is based on finite sample spaces, as it is in Algebraic Statistics (AS).
This talk will first review basic IG from the point of view of AS. In the second part, it discusses the issue of computation IG quantities and presents a few examples.

Mariya Ponomarenko (SchlumbergerDoll Research) 
IMA/MCIM Industrial problems seminar: Downhole analysis of hydrocarbons 
Abstract: Quick and accurate estimation of the composition of the hydrocarbon
fluid in the formation is essential in assessing an oil reservoir
value and determining optimal production strategies. This task is
complicated by contamination from oil and syntheticbased drilling
mud filtrates. In this talk we will describe the visible 
nearinfrared spectroscopy technique to estimate the composition of
formation fluid and level of contamination from the downhole optical
absorption spectroscopy measurements. 
Thomas S. Richardson (University of Washington) 
Gaussian path diagrams 
Abstract: In the 1920's the geneticist Sewall Wright introduced a class of
Gaussian statistical models represented by graphs containing directed
and bidirected edges, known as path diagrams. These models have been
used extensively in psychometrics and econometrics where they are
called structural equation models.
I will first describe the subclass of bowfree acyclic path diagrams, which have desirable statistical properties. I will then characterize a subclass of models that are characterized by their Markov properties. Lastly I will outline recent work aimed at characterizing nonMarkovian constraints that may arise.
(This is joint work with Mathias Drton, Michael Eichler and Masashi Miyamura.)

Daniel Robertz (RWTH Aachen) 
IMA postdoc seminar: Janet's algorithm for modules over polynomial rings 
Abstract: This talk gives an introduction to Janet bases.
Originally developed for the algebraic analysis of systems
of partial differential equations in the beginning of the
20th century, the algorithm by Maurice Janet is today an
efficient alternative for Buchberger's algorithm to compute
Gröbner bases of modules over polynomial rings.
In this talk we give a modern description of Janet's
algorithm and explain nice combinatorial properties of
the resulting Janet bases: separation of the variables into
multiplicative and nonmultiplicative ones for each Janet
basis element allows to read off vector space bases for
both the submodule and the residue class module. As a
consequence, the Hilbert series and polynomial of a (graded)
module as well as a free resolution are easily obtained
from the Janet basis.
If time admits, some modifications of Janet's algorithm
will be addressed which allow to work with polynomial rings
over the integers instead of a field resp. generalize the
algorithm to certain classes of noncommutative polynomial
rings. 
Aleksandra B. Slavković (Pennsylvania State University) 
Application of algebraic statistics for statistical disclosure
limitation 
Abstract: Statistical disclosure limitation applies statistical tools to the
problems of limiting sensitive information releases about individuals and
groups that are part of statistical databases while allowing for proper
statistical inference. The limited releases can be in a form of arbitrary
collections of marginal and conditional distributions, and odds ratios for
contingency tables. Given this information, we discuss how tools from
algebraic geometry can be used to give both complete and incomplete
characterization of discrete distributions for contingency tables. These
problems also lead to linear and nonlinear integer optimization
formulations. We discuss some practical implication, and challenges, of
using algebraic statistics for data privacy and confidentiality problems. 
Bernd Sturmfels (University of California) 
Open problems in algebraic statistics 
Abstract: This talk introduces five or six mathematical problems whose solution would likely be a significant contribution to the emerging interactions between algebraic geometry, statistics, and computational biology. 
Seth Sullivant (Harvard University) 
Algebraic geometry and applications seminar: Algebraic geometry of Gaussian Bayesian networks 
Abstract: Conditional independence models for Gaussian random variables
are algebraic varieties in the cone of positive definite matrices. We
explore the geometry of these varieties in the case of Bayesian
networks,
with a view towards generalizing the recursive factorization theorem.
When some of the random variables are hidden, nonindependence
constraints
are need to describe the Bayesian networks. These nonindependence
constraints have potential inferential uses for studying collections of
random variables. In the case that the underlying network is a tree, we
give a complete description of the defining constraints of the model and
show a surprising connection to the Grassmannian. 
Thorsten Theobald (Johann Wolfgang GoetheUniversität Frankfurt) 
Algebraic geometry and applications seminar: Symmetries in SDPbased relaxations for constrained polynomial optimization 
Abstract: We consider the issue of exploiting symmetries in the
hierarchy of semidefinite programming relaxations
recently introduced in polynomial optimization.
After providing the necessary background we focus
on problems where either the symmetric or
the cyclic group is acting on the variables
and extend the representationtheoretical
methods of Gatermann and Parrilo to constrained
polynomial optimization problems.
Moreover, we also propose methods to efficiently compute
lower and upper bounds for the subclass of problems where
the objective function and the constraints are described
in terms of power sums.
(Joint work with L. Jansson, J.B. Lasserre and C. Riener) 
Jaroslaw Wisniewski (University of Warsaw) 
On phylogenetic trees – a geometer's view 
Abstract: I will discuss geometric methods of investigating phylogenetic trees. In a joint project with Weronika Buczynska we investigate projective varieties which are binary symmetric models of trivalent phylogenetic trees. They have Gorenstein terminal singularities and are Fano varieties. Moreover any two such varieties which are of the same dimension are deformation equivalent, that is, they are in the same connected component of the Hilbert scheme of the projective space whose coordinates are indexed by subsets of their leaves. 
Ruriko Yoshida (University of Kentucky) 
A combinatorial test for significant codivergence between coolseason
grasses and their symbiotic fungal endophytes 
Abstract: Symbioses of grasses and fungal endophytes constitute an
interesting model
for evolution of mutualism and parasitism. Grasses of all
subfamilies can
harbor systemic infections by fungi of the family
Clavicipitaceae. Subfamily
Poöideae is specifically associated with epichloë
endophytes (species
of Epichloë and their asexual derivatives, the
Neotyphodium
species) in
intimate symbioses often characterized by highly efficient
vertical
transmission in seeds, and bioprotective benefits conferred by
the
symbionts to their hosts. These remarkable symbioses have been
identified
in most grass tribes spanning the taxonomic range of the
subfamily. Here we
examine the possibility of codivergence in the phylogenetic
histories of
Poöideae and epichloë. We introduce a method of analysis to
detect significant codivergence even in the absence of strict
cospeciation, and to address problems in previously developed methods. Relative ages
of
corresponding cladogenesis events were determined from
ultrametric maximum
likelihood H (host) and P (parasite = symbiont) trees by
an algorithm
called MRCALink (most recent common ancestor link), an
improvement over
previous methods that greatly weight deep over shallow H and
P node
pairs.
We then compared the inferred correspondence of MRCA ages in
the H and
P trees to the spaces of trees estimated from 10,000 randomly
generated
H and P tree pairs. Analysis of the complete dataset, which
included a
broad hostrange species and some likely host transfers
(jumps), did not
indicate significant codivergence. However, when likely host
jumps were
removed the analysis indicated highly significant codivergence.
Interestingly,
early cladogenesis events in the Poöideae corresponded to
early
cladogenesis
events in epichloë, suggesting concomitant origins of the
Poöideae and
this unusual symbiotic system.
This is joint work with C. L. Schardl, K. D. Craven, A.
Lindstrom, and
A. Stromberg. 
Debbie Yuster (Columbia University) 
Classifying disease models using regular polyhedral subdivisions 
Abstract: Genes play a complicated role in how likely one is to get a certain disease. Biologists would like to model how one's genotype affects their likelihood of illness. We propose a new classification of twolocus disease models, where each model corresponds to an induced subdivision of a point configuration (basically a picture of connected dots). Our models reflect epistasis, or gene interaction. This work is joint with Ingileif Hallgrimsdottir. For more information, see our preprint at arXiv:qbio.QM/0612044. 
Yi Zhou (CarnegieMellon University) 
Maximum likelihood estimation in latent class 
Abstract: Latent class models have been used to explain the heterogeneity of the observed relationship among a set of categorical variables and have received more and more attention as a powerful methodology for analyzing discrete data. The central goal of our work is to study the existence and computation of maximum likelihood estimates (MLEs) for these models, which are cardinal for assessment of goodness of fit and model selection. Our study is at the interface between the fields of algebraic statistics and machine learning.
Traditionally, the expectation maximization (EM) algorithm has been applied to compute the MLEs of a latent class model. However, the solutions provided by the EM correspond to local maxima only, so, although we are able to compute them effectively, we still lack methods for assessing uniqueness and existence of the MLEs. Another interesting problem in statistics is the identifiability of the model. When a model is unidentifiable, it is necessary to adjust the number of degrees of freedom in order to apply correctly goodnessoffit tests. In our work, we show that both the existence and identifiability problems are closely related to the geometric properties of the latent class models. Therefore, studying the algebraic varieties and ideals arising from these models is particularly relevant to our problem. We include a number of examples as a way of opening a discussion on a general method for addressing both MLE existence and identifiability in latent class models.
