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Nanny Wermuth

Publications and source records attributed to Nanny Wermuth.

14 recordsLinked to original sources
arXiv2017

Generating large Ising models with Markov structure via simple linear relations

We extend the notion of a tree graph to sequences of prime graphs which are cycles and edges and name these non-chordal graphs hollow trees. These structures are especially attractive for palindromic Ising models, which mimic a symmetry of joint Gaussian distributions. We show that for an Ising model all defining independences are captured by zero partial correlations and conditional correlations agree with partial correlations within each prime graph if and only if the model is palindromic and has a hollow-tree structure. This implies that the strength of dependences can be assessed locally. We use the results to find a well-fitting general Ising model with hollow-tree structure for a set of longitudinal data.

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arXiv2017

Pairwise Markov properties for regression graphs

With a sequence of regressions, one may generate joint probability distributions. One starts with a joint, marginal distribution of context variables having possibly a concentration graph structure and continues with an ordered sequence of conditional distributions, named regressions in joint responses. The involved random variables may be discrete, continuous or of both types. Such a generating process specifies for each response a conditioning set which contains just its regressor variables and it leads to at least one valid ordering of all nodes in the corresponding regression graph which has three types of edge; one for undirected dependences among context variables, another for undirected dependences among joint responses and one for any directed dependence of a response on a regressor variable. For this regression graph, there are several definitions of pairwise Markov properties, where each interprets the conditional independence associated with a missing edge in the graph in a different way. We explain how these properties arise, prove their equivalence for compositional graphoids and point at the equivalence of each one of them to the global Markov property.

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arXiv2016

Palindromic Bernoulli distributions

We introduce and study a subclass of joint Bernoulli distributions which has the palindromic property. For such distributions the vector of joint probabilities is unchanged when the order of the elements is reversed. We prove for binary variables that the palindromic property is equivalent to zero constraints on all odd-order interaction parameters, be it in parameterizations which are log-linear, linear or multivariate logistic. In particular, we derive the one-to-one parametric transformations for these three types of model specifications and give simple closed forms of maximum likelihood estimates. Some special cases and a case study are described.

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arXiv2016

Total positivity in Markov structures

We discuss properties of distributions that are multivariate totally positive of order two (MTP2) related to conditional independence. In particular, we show that any independence model generated by an MTP2 distribution is a compositional semigraphoid which is upward-stable and singleton-transitive. In addition, we prove that any MTP2 distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of MTP2 distributions and discuss ways of constructing MTP2 distributions; in particular we give conditions on the log-linear parameters of a discrete distribution which ensure MTP2 and characterize conditional Gaussian distributions which satisfy MTP2.

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arXiv2015

Graphical Markov models, unifying results and their interpretation

Graphical Markov models combine conditional independence constraints with graphical representations of stepwise data generating processes.The models started to be formulated about 40 years ago and vigorous development is ongoing. Longitudinal observational studies as well as intervention studies are best modeled via a subclass called regression graph models and, especially traceable regressions. Regression graphs include two types of undirected graph and directed acyclic graphs in ordered sequences of joint responses. Response components may correspond to discrete or continuous random variables and may depend exclusively on variables which have been generated earlier. These aspects are essential when causal hypothesis are the motivation for the planning of empirical studies. To turn the graphs into useful tools for tracing developmental pathways and for predicting structure in alternative models, the generated distributions have to mimic some properties of joint Gaussian distributions. Here, relevant results concerning these aspects are spelled out and illustrated by examples. With regression graph models, it becomes feasible, for the first time, to derive structural effects of (1) ignoring some of the variables, of (2) selecting subpopulations via fixed levels of some other variables or of (3) changing the order in which the variables might get generated. Thus, the most important future applications of these models will aim at the best possible integration of knowledge from related studies.

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arXiv2015

Graphical Markov models: overview

We describe how graphical Markov models started to emerge in the last 40 years, based on three essential concepts that had been developed independently more than a century ago. Sequences of joint or single regressions and their regression graphs are singled out as being best suited for analyzing longitudinal data and for tracing developmental pathways. Interpretations are illustrated using two sets of data and some of the more recent, important results for sequences of regressions are summarized.

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arXiv2014

Star graphs induce tetrad correlations: for Gaussian as well as for binary variables

Tetrad correlations were obtained historically for Gaussian distributions when tasks are designed to measure an ability or attitude so that a single unobserved variable may generate the observed, linearly increasing dependences among the tasks. We connect such generating processes to a particular type of directed graph, the star graph, and to the notion of traceable regressions. Tetrad correlation conditions for the existence of a single latent variable are derived. These are needed for positive dependences not only in joint Gaussian but also in joint binary distributions. Three applications with binary items are given.

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arXiv2013

Concepts and a case study for a flexible class of graphical Markov models

With graphical Markov models, one can investigate complex dependences, summarize some results of statistical analyses with graphs and use these graphs to understand implications of well-fitting models. The models have a rich history and form an area that has been intensively studied and developed in recent years. We give a brief review of the main concepts and describe in more detail a flexible subclass of models, called traceable regressions. These are sequences of joint response regressions for which regression graphs permit one to trace and thereby understand pathways of dependence. We use these methods to reanalyze and interpret data from a prospective study of child development, now known as the Mannheim Study of Children at Risk. The two related primary features concern cognitive and motor development, at the age of 4.5 and 8 years of a child. Deficits in these features form a sequence of joint responses. Several possible risks are assessed at birth of the child and when the child reached age 3 months and 2 years.

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arXiv2012

Sequences of regressions and their dependences

In this paper, we define and study the concept of traceable regressions. These are sequences of regressions in joint or single responses for which a corresponding regression graph captures not only an independence structure but represents, in addition, conditional dependences that permit the tracing of pathways of dependence. We give the properties needed for transforming these graphs and graphical criteria to decide whether a path in the graph induces a dependence. The much stronger constraints on distributions that are faithful to a graph are compared to those needed for traceable regressions.

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arXiv2012

Sequences of regressions and their independences

Ordered sequences of univariate or multivariate regressions provide statistical models for analysing data from randomized, possibly sequential interventions, from cohort or multi-wave panel studies, but also from cross-sectional or retrospective studies. Conditional independences are captured by what we name regression graphs, provided the generated distribution shares some properties with a joint Gaussian distribution. Regression graphs extend purely directed, acyclic graphs by two types of undirected graph, one type for components of joint responses and the other for components of the context vector variable. We review the special features and the history of regression graphs, derive criteria to read all implied independences of a regression graph and prove criteria for Markov equivalence that is to judge whether two different graphs imply the same set of independence statements. Knowledge of Markov equivalence provides alternative interpretations of a given sequence of regressions, is essential for machine learning strategies and permits to use the simple graphical criteria of regression graphs on graphs for which the corresponding criteria are in general more complex. Under the known conditions that a Markov equivalent directed acyclic graph exists for any given regression graph, we give a polynomial time algorithm to find one such graph.

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arXiv2012

Case-control studies for rare diseases: improved estimation of several risks and of feature dependences

To capture the dependences of a disease on several risk factors, a challenge is to combine model-based estimation with evidence-based arguments. Standard case-control methods allow estimation of the dependences of a rare disease on several regressors via logistic regressions. For case-control studies, the sampling design leads to samples from two different populations and for the set of regressors in every logistic regression, these samples are then mixed and taken as given observations. But, it is the differences in independence structures of regressors for cases and for controls that can improve logistic regression estimates and guide us to the important feature dependences that are specific to the diseased. A case-control study on laryngeal cancer is used as illustration.

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arXiv2011

Probability distributions with summary graph structure

A set of independence statements may define the independence structure of interest in a family of joint probability distributions. This structure is often captured by a graph that consists of nodes representing the random variables and of edges that couple node pairs. One important class contains regression graphs. Regression graphs are a type of so-called chain graph and describe stepwise processes, in which at each step single or joint responses are generated given the relevant explanatory variables in their past. For joint densities that result after possible marginalising or conditioning, we introduce summary graphs. These graphs reflect the independence structure implied by the generating process for the reduced set of variables and they preserve the implied independences after additional marginalising and conditioning. They can identify generating dependences that remain unchanged and alert to possibly severe distortions due to direct and indirect confounding. Operators for matrix representations of graphs are used to derive these properties of summary graphs and to translate them into special types of paths in graphs.

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arXiv2009

Matrix representations and independencies in directed acyclic graphs

For a directed acyclic graph, there are two known criteria to decide whether any specific conditional independence statement is implied for all distributions factorized according to the given graph. Both criteria are based on special types of path in graphs. They are called separation criteria because independence holds whenever the conditioning set is a separating set in a graph theoretical sense. We introduce and discuss an alternative approach using binary matrix representations of graphs in which zeros indicate independence statements. A matrix condition is shown to give a new path criterion for separation and to be equivalent to each of the previous two path criteria.

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