Maximum likelihood estimation of the Latent Class Model through model boundary decomposition

The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.

Inference for Ordinal Log-Linear Models Based on Algebraic Statistics

Tools of algebraic statistics combined with MCMC algorithms have been used in contingency table analysis for model selection and model fit testing of log-linear models. However, this approach has not been considered so far for association models, which are special log-linear models for tables with ordinal classification variables. The simplest association model for two-way tables, the uniform (U) association model, has just one parameter more than the independence model and is applicable when both classification variables are ordinal. Less parsimonious are the row (R) and column (C) effect association models, appropriate when at least one of the classification variables is ordinal. Association models have been extended for multidimensional contingency tables as well. Here, we adjust algebraic methods for association models analysis and investigate their eligibility, focusing mainly on two-way tables. They are implemented in the statistical software R and illustrated on real data tables. Finally the algebraic model fit and selection procedure is assessed and compared to the asymptotic approach in terms of a simulation study.

Stephen Fienberg's influence on algebraic statistics

Stephen Fienberg (1942-2016) was a statistician whose career has been an inspiration for the engagement of statistics with social and scientific issues, and it is in this spirit that he helped steer algebraic statistics toward more of a mainstream. Many of his favorite topics in the area are covered in this special issue. We are grateful to all authors for contributing to this volume to honor him and his influence on the field. During the preparation of this issue, we also learned about the tragic killing of his widow, Joyce Fienberg, in the Tree of Life Synagogue in Pittsburgh. This issue is dedicated to their memory.

Strongly Robust Toric Ideals in Codimension 2

A homogeneous ideal is robust if its universal Gröbner basis is also a minimal generating set. For toric ideals, one has the stronger definition: A toric ideal is strongly robust if its Graver basis equals the set of indispensable binomials. We characterize the codimension 2 strongly robust toric ideals by their Gale diagrams. This give a positive answer to a question of Petrovic, Thoma, and Vladoiu in the case of codimension 2 toric ideals.

Cubature rules and expected value of some complex functions

The expected value of some complex valued random vectors is computed by means of the indicator function of a designed experiment as known in algebraic statistics. The general theory is set-up and results are obtained for nite discrete random vectors and the Gaussian random vector. The precision space of some cubature rules/designed experiments are determined.

On Exchangeability in Network Models

We derive representation theorems for exchangeable distributions on finite and infinite graphs using elementary arguments based on geometric and graph-theoretic concepts. Our results elucidate some of the key differences, and their implications, between statistical network models that are finitely exchangeable and models that define a consistent sequence of probability distributions on graphs of increasing size.

Exact tests to compare contingency tables under quasi-independence and quasi-symmetry

In this work we define log-linear models to compare several square contingency tables under the quasi-independence or the quasi-symmetry model, and the relevant Markov bases are theoretically characterized. Through Markov bases, an exact test to evaluate if two or more tables fit a common model is introduced. Two real-data examples illustrate the use of these models in different fields of applications.

Generalized Fréchet Bounds for Cell Entries in Multidimensional Contingency Tables

We consider the lattice, $\mathcal{L}$, of all subsets of a multidimensional contingency table and establish the properties of monotonicity and supermodularity for the marginalization function, $n(\cdot)$, on $\mathcal{L}$. We derive from the supermodularity of $n(\cdot)$ some generalized Fr\'echet inequalities complementing and extending inequalities of Dobra and Fienberg. Further, we construct new monotonic and supermodular functions from $n(\cdot)$, and we remark on the connection between supermodularity and some correlation inequalities for probability distributions on lattices. We also apply an inequality of Ky Fan to derive a new approach to Fr\'echet inequalities for multidimensional contingency tables.

Mixtures and products in two graphical models

We compare two statistical models of three binary random variables. One is a mixture model and the other is a product of mixtures model called a restricted Boltzmann machine. Although the two models we study look different from their parametrizations, we show that they represent the same set of distributions on the interior of the probability simplex, and are equal up to closure. We give a semi-algebraic description of the model in terms of six binomial inequalities and obtain closed form expressions for the maximum likelihood estimates. We briefly discuss extensions to larger models.

Maximal Length Projections in Group Algebras with Applications to Linear Rank Tests of Uniformity

Let \(G\) be a finite group, let \(\mathbb{C}G\) be the complex group algebra of \(G\), and let \(p \in \mathbb{C}G\). In this paper, we show how to construct submodules\(S\) of \(\mathbb{C}G\) of a fixed dimension with the property that the orthogonal projection of \(p\) onto \(S\) has maximal length. We then provide an example of how such submodules for the symmetric group \(S_n\) can be used to create new linear rank tests of uniformity in statistics for survey data that arises when respondents are asked to give a complete ranking of \(n\) items.