On the Markov Equivalence of Chain Graphs, Undirected Graphs, and Acyclic Digraphs

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.     

Parameterizations and Fitting of Bi-directed Graph Models to Categorical Data

Abstract.  We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as thenation multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation-independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and Silvey method.     

Estimation of General Stationary Processes by Variable Length Markov Chains

We develop new results about a sieve methodology for the estimation of minimal state spaces and probability laws in the class of stationary processes defined on finite categorical spaces. Using a sieve approximation with variable length Markov chains of increasing order, we show that an adapted version of the Context algorithm yields asymptotically correct estimates for the minimal state space and for the underlying probability distribution. As a side product, the method of sieves yields a nice graphical tree representation for the potentially infinite dimensional minimal state space of the data generating process, which is very useful for exploration of the memory.     

A Unified Approach to the Characterization of Equivalence Classes of DAGs, Chain Graphs with no Flags and Chain Graphs

Abstract.  A Markov property associates a set of conditional independencies to a graph. Two alternative Markov properties are available for chain graphs (CGs), the Lauritzen–Wermuth–Frydenberg (LWF) and the Andersson–Madigan– Perlman (AMP) Markov properties, which are different in general but coincide for the subclass of CGs with no flags . Markov equivalence induces a partition of the class of CGs into equivalence classes and every equivalence class contains a, possibly empty, subclass of CGs with no flags itself containing a, possibly empty, subclass of directed acyclic graphs (DAGs). LWF-Markov equivalence classes of CGs can be naturally characterized by means of the so-called largest CGs , whereas a graphical characterization of equivalence classes of DAGs is provided by the essential graphs . In this paper, we show the existence of largest CGs with no flags that provide a natural characterization of equivalence classes of CGs of this kind, with respect to both the LWF- and the AMP-Markov properties. We propose a procedure for the construction of the largest CGs, the largest CGs with no flags and the essential graphs, thereby providing a unified approach to the problem. As by-products we obtain a characterization of graphs that are largest CGs with no flags and an alternative characterization of graphs which are largest CGs. Furthermore, a known characterization of the essential graphs is shown to be a special case of our more general framework. The three graphical characterizations have a common structure: they use two versions of a locally verifiable graphical rule. Moreover, in case of DAGs, an immediate comparison of three characterizing graphs is possible.     

C219. Comment on ”calculating binomial probabilities when the trial probabilities are unequal“

In complex models like hidden Markov chains, the convergence of the MCMC algorithms used to approximate the posterior distribution and the Bayes estimates of the parameters of interest must be controlled in a robust manner. We propose in this paper a series of online controls, which rely on classical non-parametric tests, to evaluate independence from the start-up distribution, stability of the Markov chain, and asymptotic normality. These tests lead to graphical control spreadsheets which arepresentedin the set-up of normalmixture hidden Markov chains to compare the full Gibbs sampler with an aggregated Gibbs sampler based on the forward – backward formulas.     

Quasi‐Symmetric Graphical Log‐Linear Models

Abstract. We propose an extension of graphical log‐linear models to allow for symmetry constraints on some interaction parameters that represent homologous factors. The conditional independence structure of such quasi‐symmetric (QS) graphical models is described by an undirected graph with coloured edges, in which a particular colour corresponds to a set of equality constraints on a set of parameters. Unlike standard QS models, the proposed models apply with contingency tables for which only some variables or sets of the variables have the same categories. We study the graphical properties of such models, including conditions for decomposition of model parameters and of maximum likelihood estimates.     

Bayesian model-averaged regularization for Gaussian graphical models

The graphical lasso has now become a useful tool to estimate high-dimensional Gaussian graphical models, but its practical applications suffer from the problem of choosing regularization parameters in a data-dependent way. In this article, we propose a model-averaged method for estimating sparse inverse covariance matrices for Gaussian graphical models. We consider the graphical lasso regularization path as the model space for Bayesian model averaging and use Markov chain Monte Carlo techniques for the regularization path point selection. Numerical performance of our method is investigated using both simulated and real datasets, in comparison with some state-of-art model selection procedures.     

Piecewise deterministic Markov processes applied to fatigue crack growth modelling

In this paper, we use a particular piecewise deterministic Markov process (PDMP) to model the evolution of a degradation mechanism that may arise in various structural components, namely, the fatigue crack growth. We first derive some probability results on the stochastic dynamics with the help of Markov renewal theory: a closed-form solution for the transition function of the PDMP is given. Then, we investigate some methods to estimate the parameters of the dynamical system, involving Bogolyubov's averaging principle and maximum likelihood estimation for the infinitesimal generator of the underlying jump Markov process. Numerical applications on a real crack data set are given.     

Level-Phase Independence for Fluid Queues

ABSTRACT

A Markov-modulated fluid queue is a two-dimensional Markov process; the first dimension is continuous and is usually called the level, and the second is the state of a Markov process that determines the evolution of the level, it is usually called the phase. We show that it is always possible to modify the transition rules at the boundary level of the fluid queue in order to obtain independence between the level and the phase under the stationary distribution. We obtain this result by exploiting the similarity between fluid queues and Quasi-Birth-and-Death (QBD) processes.     

Generic reversible jump MCMC using graphical models

Markov chain Monte Carlo techniques have revolutionized the field of Bayesian statistics. Their power is so great that they can even accommodate situations in which the structure of the statistical model itself is uncertain. However, the analysis of such trans-dimensional (TD) models is not easy and available software may lack the flexibility required for dealing with the complexities of real data, often because it does not allow the TD model to be simply part of some bigger model. In this paper we describe a class of widely applicable TD models that can be represented by a generic graphical model, which may be incorporated into arbitrary other graphical structures without significantly affecting the mechanism of inference. We also present a decomposition of the reversible jump algorithm into abstract and problem-specific components, which provides infrastructure for applying the method to all models in the class considered. These developments represent a first step towards a context-free method for implementing TD models that will facilitate their use by applied scientists for the practical exploration of model uncertainty. Our approach makes use of the popular WinBUGS framework as a sampling engine and we illustrate its use via two simple examples in which model uncertainty is a key feature.     

Label‐ and Level‐Invariant Graphical Log‐Linear Models

We introduce two types of graphical log‐linear models: label‐ and level‐invariant models for triangle‐free graphs. These models generalise symmetry concepts in graphical log‐linear models and provide a tool with which to model symmetry in the discrete case. A label‐invariant model is category‐invariant and is preserved after permuting some of the vertices according to transformations that maintain the graph, whereas a level‐invariant model equates expected frequencies according to a given set of permutations. These new models can both be seen as instances of a new type of graphical log‐linear model termed the restricted graphical log‐linear model, or RGLL, in which equality restrictions on subsets of main effects and first‐order interactions are imposed. Their likelihood equations and graphical representation can be obtained from those derived for the RGLL models.     

Additive mixed models with approximate Dirichlet process mixtures: the EM approach

We consider additive mixed models for longitudinal data with a nonlinear time trend. As random effects distribution an approximate Dirichlet process mixture is proposed that is based on the truncated version of the stick breaking presentation of the Dirichlet process and provides a Gaussian mixture with a data driven choice of the number of mixture components. The main advantage of the specification is its ability to identify clusters of subjects with a similar random effects structure. For the estimation of the trend curve the mixed model representation of penalized splines is used. An Expectation-Maximization algorithm is given that solves the estimation problem and that exhibits advantages over Markov chain Monte Carlo approaches, which are typically used when modeling with Dirichlet processes. The method is evaluated in a simulation study and applied to theophylline data and to body mass index profiles of children.     

Generalized order statistic processes and Pfeifer records

We introduce a uniform generalized order statistic process. It is a simple Markov process whose initial segment can be identified with a set of uniform generalized order statistics. A standard marginal transformation leads to a generalized order statistic process related to non-uniform generalized order statistics. It is then demonstrated that the nth variable in such a process has the same distribution as an nth Pfeifer record value. This process representation of Pfeifer records facilitates discussion of the possible limit laws for Pfeifer records and, in some cases, of sums thereof. Because of the close relationship between Pfeifer records and generalized order statistics, the results shed some light on the problem of determining the nature of the possible limiting distributions of the largest generalized order statistic.     

On Block Ordering of Variables in Graphical Modelling

Abstract.  In graphical modelling, the existence of substantive background knowledge on block ordering of variables is used to perform structural learning within the family of chain graphs (CGs) in which every block corresponds to an undirected graph and edges joining vertices in different blocks are directed in accordance with the ordering. We show that this practice may lead to an inappropriate restriction of the search space and introduce the concept of labelled block ordering B corresponding to a family of B - consistent CGs in which every block may be either an undirected graph or a directed acyclic graph or, more generally, a CG. In this way we provide a flexible tool for specifying subsets of chain graphs, and we observe that the most relevant subsets of CGs considered in the literature are families of B -consistent CGs for the appropriate choice of B . Structural learning within a family of B -consistent CGs requires to deal with Markov equivalence. We provide a graphical characterization of equivalence classes of B -consistent CGs, namely the B - essential graphs , as well as a procedure to construct the B -essential graph for any given equivalence class of B -consistent chain graphs. Both largest CGs and essential graphs turn out to be special cases of B -essential graphs.     

A New Approach to Volatility Modeling: The Factorial Hidden Markov Volatility Model

A new process—the factorial hidden Markov volatility (FHMV) model—is proposed to model financial returns or realized variances. Its dynamics are driven by a latent volatility process specified as a product of three components: a Markov chain controlling volatility persistence, an independent discrete process capable of generating jumps in the volatility, and a predictable (data-driven) process capturing the leverage effect. An economic interpretation is attached to each one of these components. Moreover, the Markov chain and jump components allow volatility to switch abruptly between thousands of states, and the transition matrix of the model is structured to generate a high degree of volatility persistence. An empirical study on six financial time series shows that the FHMV process compares favorably to state-of-the-art volatility models in terms of in-sample fit and out-of-sample forecasting performance over time horizons ranging from 1 to 100 days. Supplementary materials for this article are available online.     

On the multivariate predictive distribution of multi-dimensional effective dose: a Bayesian approach

We propose a Bayesian procedure to sample from the distribution of the multi-dimensional effective dose. This effective dose is the set of dose levels of multiple predictive factors that produce a binary response with a fixed probability.We apply our algorithms to parametric and semiparametric logistics regression models, respectively. The graphical display of random samples obtained through Markov chain Monte Carlo can provide some insight into the predictive distribution.     

Structured priors for multivariate time series

A class of prior distributions for multivariate autoregressive models is presented. This class of priors is built taking into account the latent component structure that characterizes a collection of autoregressive processes. In particular, the state-space representation of a vector autoregressive process leads to the decomposition of each time series in the multivariate process into simple underlying components. These components may have a common structure across the series. A key feature of the proposed priors is that they allow the modeling of such common structure. This approach also takes into account the uncertainty in the number of latent processes, consequently handling model order uncertainty in the multivariate autoregressive framework. Posterior inference is achieved via standard Markov chain Monte Carlo (MCMC) methods. Issues related to inference and exploration of the posterior distribution are discussed. We illustrate the methodology analyzing two data sets: a synthetic data set with quasi-periodic latent structure, and seasonally adjusted US monthly housing data consisting of housing starts and housing sales over the period 1965 to 1974.     

A graphical model selection tool for mixed models

ABSTRACT

Model selection can be defined as the task of estimating the performance of different models in order to choose the most parsimonious one, among a potentially very large set of candidate statistical models. We propose a graphical representation to be considered as an extension to the class of mixed models of the deviance plot proposed in the literature within the framework of classical and generalized linear models. This graphical representation allows, once a reduced number of models have been selected, to identify important covariates focusing only on the fixed effects component, assuming the random part properly specified. Nevertheless, we suggest also a standalone figure representing the residual random variance ratio: a cross-evaluation of the two graphical representations will allow to derive some conclusions on the random part specification of the model and a more accurate selection of the final model.     

Stratified Gaussian graphical models

Gaussian graphical models represent the backbone of the statistical toolbox for analyzing continuous multivariate systems. However, due to the intrinsic properties of the multivariate normal distribution, use of this model family may hide certain forms of context-specific independence that are natural to consider from an applied perspective. Such independencies have been earlier introduced to generalize discrete graphical models and Bayesian networks into more flexible model families. Here, we adapt the idea of context-specific independence to Gaussian graphical models by introducing a stratification of the Euclidean space such that a conditional independence may hold in certain segments but be absent elsewhere. It is shown that the stratified models define a curved exponential family, which retains considerable tractability for parameter estimation and model selection.     

Statistical Inference in Context Specific Interaction Models for Contingency Tables

Abstract.  Context specific interaction models is a class of interaction models for contingency tables in which interaction terms are allowed to vanish in specific contexts given by the levels of sets of variables. Such restrictions can entail conditional independencies which only hold for some values of the conditioning variables and allows also for irrelevance of some variables in specific contexts. A Markov property is established and so is an iterative proportional scaling algorithm for maximum likelihood estimation. Decomposition of the estimation problem is treated and model selection is discussed.