We consider the problem of model selection and accounting for model uncertainty in high-dimensional contingency tables, motivated by expert system applications. The approach most used currently is a stepwise strategy guided by tests based on approximate asymptotic P-values leading to the selection of a single model; inference is then conditional on the selected model. The sampling properties of such a strategy are complex, and the failure to take account of model uncertainty leads to underestimation of uncertainty about quantities of interest. In principle, a panacea is provided by the standard Bayesian formalism which averages the posterior distributions of the quantity of interest under each of the models, weighted by their posterior model probabilities. Furthermore, this approach is optimal in the sense of maximising predictive ability. However, this has not been used in practice because computing the posterior model probabilities is hard and the number of models is very large (often greater than 1011). We argue that the standard Bayesian formalism is unsatisfactory and we propose an alternative Bayesian approach that, we contend, takes full account of the true model uncertainty byaveraging overamuch smaller set of models. An efficient search algorithm is developed for nding these models. We consider two classes of graphical models that arise in expert systems: the recursive causal models and the decomposable

Undirected graphs and acyclic digraphs (ADGs), as well as their mutual extension to chain graphs, are widely used to describe dependencies among variables in multivariate distributions. In particular, the likelihood functions of ADG models admit convenient recursive factorizations that often allow explicit maximum likelihood estimates and that are well suited to building Bayesian networks for expert systems. Whereas the undirected graph associated with a dependence model is uniquely determined, there may, however, be many ADGs that determine the same dependence ( = Markov) model. Thus, the family of all ADGs with a given set of vertices is naturally partitioned into Markov-equivalence classes, each class being associated with a unique statistical model. Statistical procedures, such as model selection or model averaging, that fail to take into account these equivalence classes, may incur substantial computational or other inefficiencies. Here it is shown that each Markov-equivalence class is uniquely determined by a single chain graph, the essential graph, that is itself simultaneously Markov equivalent to all ADGs in the equivalence class. Essential graphs are characterized, a polynomial-time algorithm for their construction is given, and their applications to model selection and other statistical

Graphical Markov models use undirected graphs (UDGs), acyclic directed graphs (ADGs), or (mixed) chain graphs to represent possible dependencies among random variables in a multivariate distribution. Whereas a UDG is uniquely determined by its associated Markov model, this is not true for ADGs or for general chain graphs (which include both UDGs and ADGs as special cases). This paper addresses three questions regarding the equivalence of graphical Markov models: when is a given chain graph Markov equivalent (1) to some UDG? (2) to some (at least one) ADG? (3) to some decomposable UDG? The answers are obtained by means of an extension of Frydenberg's (1990) elegant graph-theoretic characterization of the Markov equivalence of chain graphs. 1 Introduction The use of graphs to represent dependence relations among random variables, first introduced by Wright (1921), has generated considerable research activity, especially since the early 1980s. Particular attention has been devoted to gra...

A Bayesian approach to model selection for structural equation models is outlined. This enables us to compare individual models, nested or non-nested, and also to search through the (perhaps vast) set of possible models for the best ones. The approach selects several models rather than just one, when appropriate, and so enables us to take account, both informally and formally, of uncertainty about model structure when making inferences about quantities of interest. The approach tends to select simpler models than strategies based on multiple P-value-based tests. It may thus help to overcome the criticism of structural

The theoretical background for a program for establishing expert systems on the basis of observations and expert knowledge is presented. Block recursive models form the basis of the statistical modelling. These models, together with various model selection methods for automatic model selection, are presented. Additionally, the connection between a block recursive model and expert systems based on causal probabilistic networks is treated. A medical example concerning diagnosis of coronary artery disease forms the basis for an evaluation of the expert systems established. Keywords: causal probabilistic networks, graphical association models, machine learning, model selection, selection criteria, selection strategies. 1 Introduction BIFROST is a program for semi-automatic knowledge acquisition and is a continuation developments made in (Greve, Hjsgaard, Skjth and Thiesson 1990). The objective is to obtain preliminary causal models for use in the HUGIN expert system shell (Ander...

This research deals with experiments or surveys producing multivariate categorical data which is incomplete, in the sense that not all variables of interest are measured on every subject or element of the sample. For the most part, incompleteness is taken to arise by design, rather than by random failure of the measurement process. In these circumstances, one can often assume that counts derived from appropriate disjoint subsets of the data arise from independent multinomial distributions with linearly related parameters. Best asymptotically normal oJ estimates of these parameters may be determined by maximizing the likelihood of the observations or by minimizing Pearson's-x 2, Neyman's X~,

This paper was written in 1995/1996. The German version was published by ZA Info-mation. The English version was submitted but not revised. 1 This paper presents a general approach to the analysis of categorical panel data which is based on using causal log-linear models with latent variables. Like the well-known LISREL model, these models consist of a structural and a measurement part. In the structural part, a system of logit equations is used to explain changes which occur in the dependent variable of interest. An unrestricted or restricted latent class model is used in the measurement part of the model. It is demonstrated that the measurement model can be used to specify discrete variants of latent trait models, such as the Rasch model and the Lord-Birnbaum model. The approach is illustrated by means of an application on youth-centrism. Several measurement models are tested for a scale which is assumed to measure youth-centrism. In addition, the influence of covariates on a person’s initial position and on the transition probabilities between time points is studied.

This paper empirically analyzes effects of gender on the public's interpretations of various activities as "work." In a special survey conducted by the Census Bureau, a series of vignettes describing hypothetical work-related situations was presented to respondents, who were asked to classify them as "work," or not. The gender of the subject described in the vignette was randomly varied. In this paper, we apply log-linear models to examine the effects of the respondent's gender and the gender of the subject of the vignette upon classifications of the vignette activities as work. We conclude that (1) male respondents are more influenced by the gender of the vignette subject than female respondents, for certain kinds of activities, such as helping activities, and (2) men are more likely than women to classify certain marginal activities as work, such as work in preparation for a business, or casual labor for a few hours. The results show that gender influences interpretations of the mean...

OF THE DISSERTATION Model Checking for Incomplete High Dimensional Categorical Data by Ming-Yi Hu Doctor of Philosophy in Statistics University of California, Los Angeles, 1999 Professor Thomas R. Belin, Co-chair Professor Robert I. Jennrich, Co-chair Categorical data are often arranged in a contingency table and summarized by a loglinear model. A standard approach for comparing two competing models is to calculate twice the discrepancy between maximized loglikelihoods, which follows a 2 distribution asymptotically. But when data are sparse, the 2 approximation may be questionable. xii As an alternative to a large-sample approximation to the reference distribution, we implement the framework introduced by Rubin (1984) for finding the posterior predictive check (PPC) distribution. The PPC distribution represents the conditional probability of a future value of a test statistic based on the information given by observed data along with model specifications, which can se...

Many research designs in experimental psychology generate data that are fundamentally discrete or categorical in nature, and produce multiway tables of frequencies. Despite an extensive and, more recently, accessible literature on the topic, multiway frequency analysis is rarely used in experimental psychology. A reason may be the form of exposition in the literature, with emphases and concerns far removed from those of the typical experimental psychologist. An approach to multiway frequency analysis for experimental psychologists is described that has the features we want: asymmetrical designs, factors assessed for their respective main and interactive e#ects in a manner analogous to ANOVA, and the ability to handle within-subject designs.