Approximate Bayesian Computation for Exponential Random Graph Models for Large Social Networks

We consider the issue of sampling from the posterior distribution of exponential random graph (ERG) models and other statistical models with intractable normalizing constants. Existing methods based on exact sampling are either infeasible or require very long computing time. We study a class of approximate Markov chain Monte Carlo (MCMC) sampling schemes that deal with this issue. We also develop a new Metropolis–Hastings kernel to sample sparse large networks from ERG models. We illustrate the proposed methods on several examples.     

Exact likelihood inference for Laplace distribution based on Type-II censored samples

We develop exact inference for the location and scale parameters of the Laplace (double exponential) distribution based on their maximum likelihood estimators from a Type-II censored sample. Based on some pivotal quantities, exact confidence intervals and tests of hypotheses are constructed. Upon conditioning first on the number of observations that are below the population median, exact distributions of the pivotal quantities are expressed as mixtures of linear combinations and of ratios of linear combinations of standard exponential random variables, which facilitates the computation of quantiles of these pivotal quantities. Tables of quantiles are presented for the complete sample case.     

Density Approximation by Summary Statistics: An Information-theoretic Approach

In the case of exponential families, it is a straightforward matter to approximate a density function by use of summary statistics; however, an appropriate approach to such approximation is far less clear when an exponential family is not assumed. In this paper, a maximin argument based on information theory is used to derive a new approach to density approximation from summary statistics which is not restricted by the assumption of validity of an underlying exponential family. Information-theoretic criteria are developed to assess loss of predictive power of summary statistics under such minimal knowledge. Under these criteria, optimal density approximations in the maximin sense are obtained and shown to be related to exponential families. Conditions for existence of optimal density approximations are developed. Applications of the proposed approach are illustrated, and methods for estimation of densities are provided in the case of simple random sampling. Large-sample theory for estimates is developed.     

Exponential inequalities for unbounded functions of geometrically ergodic Markov chains: applications to quantitative error bounds for regenerative Metropolis algorithms

ABSTRACT

The aim of this note is to investigate the concentration properties of unbounded functions of geometrically ergodic Markov chains. We derive concentration properties of centred functions with respect to the square of Lyapunov's function in the drift condition satisfied by the Markov chain. We apply the new exponential inequalities to derive confidence intervals for Markov Chain Monte Carlo algorithms. Quantitative error bounds are provided for the regenerative Metropolis algorithm of [Brockwell and Kadane Identification of regeneration times in MCMC simulation, with application to adaptive schemes. J Comput Graphical Stat. 2005;14(2)].     

Stochastic comparisons of sample spacings in single- and multiple-outlier exponential models

This article studies some ordering results for the sample spacings arising from the single- and multiple-outlier exponential models. In the single-outlier exponential models, it is shown that the weak majorization order between the two hazard rate vectors implies the hazard rate order as well as the dispersive order between the corresponding sample spacings. We also extend this result from the single-outlier model to the multiple-outlier model for the special case of the second sample spacing. Furthermore, we obtain some necessary and sufficient conditions such that, on the one hand, the hazard rate, dispersive and usual stochastic orders, and on the other hand, the likelihood ratio and reversed hazard rate orders of the second sample spacings from two independent heterogeneous exponential random variables are equivalent.     

Phase Type Distributions with Finite Support

This research is motivated by the fact that many random variables of practical interest have a finite support. For fixed a < b, we consider the distribution of a random variable X = (a + Ymod(b ? a)), where Y is a phase type (PH) random variable. We demonstrate that as we traverse for Y the entire set of PH distributions (or even any subset thereof like Coxian that is dense in the class of distributions on [0, ∞)), we obtain a class of matrix exponential distributions dense in (a, b). We call these Finite Support Phase Type Distributions (FSPH) of the first kind. A simple example shows that though dense, this class by itself is not very efficient for modeling; therefore, we introduce (and derive the EM algorithms for) two other classes of finite support phase type distributions (FSPH). The properties of denseness, connection to Markov chains, the EM algorithm, and ability to exploit matrix-based computations should all make these classes of distributions attractive not only for applied probability but also for a much wider variety of fields using statistical methodologies.     

Centered parameterizations and dependence limitations in Markov random field models

Markov random field models incorporate terms representing local statistical dependence among variables in a discrete-index random field. Traditional parameterizations for models based on one-parameter exponential family conditional distributions contain components that would appear to reflect large-scale and small-scale model behaviors, and it is natural to attempt to match these structures with large-scale and small-scale patterns in a set of data. Traditional manners of parameterizing Markov random field models do not allow such correspondence, however. We propose an alternative centered parameterization that, while not leading to different models, allows a correspondence between model structures and data structures to be successfully accomplished. The ability to make these connections is important when incorporating covariate information into a model or if a sequence of models is fit over time to investigate and interpret possible changes in data structure. We demonstrate the improved interpretation that results from use of centered parameterizations. Centered parameterizations also lend themselves to computation of an interpretable decomposition of mean squared error, and this is demonstrated both analytically and through a simulated example. A breakdown in model behavior occurs even with centered parameterizations if dependence parameters in Markov random field models are allowed to become too large. This phenomenon is discussed and illustrated using an auto-logistic model.     

Markov chain Monte Carlo algorithms with sequential proposals

We explore a general framework in Markov chain Monte Carlo (MCMC) sampling where sequential proposals are tried as a candidate for the next state of the Markov chain. This sequential-proposal framework can be applied to various existing MCMC methods, including Metropolis–Hastings algorithms using random proposals and methods that use deterministic proposals such as Hamiltonian Monte Carlo (HMC) or the bouncy particle sampler. Sequential-proposal MCMC methods construct the same Markov chains as those constructed by the delayed rejection method under certain circumstances. In the context of HMC, the sequential-proposal approach has been proposed as extra chance generalized hybrid Monte Carlo (XCGHMC). We develop two novel methods in which the trajectories leading to proposals in HMC are automatically tuned to avoid doubling back, as in the No-U-Turn sampler (NUTS). The numerical efficiency of these new methods compare favorably to the NUTS. We additionally show that the sequential-proposal bouncy particle sampler enables the constructed Markov chain to pass through regions of low target density and thus facilitates better mixing of the chain when the target density is multimodal.

     

ESTIMATING WITHIN-SCHOOL CONTACT NETWORKS TO UNDERSTAND INFLUENZA TRANSMISSION

Many epidemic models approximate social contact behavior by assuming random mixing within mixing groups (e.g., homes, schools, and workplaces). The effect of more realistic social network structure on estimates of epidemic parameters is an open area of exploration. We develop a detailed statistical model to estimate the social contact network within a high school using friendship network data and a survey of contact behavior. Our contact network model includes classroom structure, longer durations of contacts to friends than non-friends and more frequent contacts with friends, based on reports in the contact survey. We performed simulation studies to explore which network structures are relevant to influenza transmission. These studies yield two key findings. First, we found that the friendship network structure important to the transmission process can be adequately represented by a dyad-independent exponential random graph model (ERGM). This means that individual-level sampled data is sufficient to characterize the entire friendship network. Second, we found that contact behavior was adequately represented by a static rather than dynamic contact network. We then compare a targeted antiviral prophylaxis intervention strategy and a grade closure intervention strategy under random mixing and network-based mixing. We find that random mixing overestimates the effect of targeted antiviral prophylaxis on the probability of an epidemic when the probability of transmission in 10 minutes of contact is less than 0.004 and underestimates it when this transmission probability is greater than 0.004. We found the same pattern for the final size of an epidemic, with a threshold transmission probability of 0.005. We also find random mixing overestimates the effect of a grade closure intervention on the probability of an epidemic and final size for all transmission probabilities. Our findings have implications for policy recommendations based on models assuming random mixing, and can inform further development of network-based models.     

The definition of tree-indexed Markov chains in random environment and their existence

This paper gives the definition of tree-indexed Markov chains in random environment with discrete state space, and then studies some equivalent theorems of tree-indexed Markov chains in random environment. Finally, we give the equivalence on tree-indexed Markov chains in Markov environment and double Markov chains indexed by a tree.     

Minimum variance unbiased estimation in natural exponential families generated by stable distributions

The class of nature exponential families generated by stable distributions has been introduced in different contexts by several authors. Tweedie (1984) and Jorgensen (1987) studied this class in the context of generalized liner models and exponential dispersion models. Bar-Lev and Enis (1986) introduced this class in the context of the property of reproducibility in natural exponential families and Hougaard (1986) found the distributions in this class to be natural candidates for applications as survival distributions in life tables for heterogeneous populations. In this note, we consider such a class in the context of minimum variance unbiased estimation. For each family in this class, we obtain an explicit expression for the uniformly minimum variance unbiased estimator for the r-th cumlant, the density function, and the reliability function.     

The double chain markov model

In the class of discrete time Markovian processes, two models are widely used, the Markov chain and the hidden Markov model. A major difference between these two models lies in the relation between successive outputs of the observed variable. In a visible Markov chain, these are directly correlated while in hidden models they are not. However, in some situations it is possible to observe both a hidden Markov chain and a direct relation between successive observed outputs. Unfortunately, the use of either a visible or a hidden model implies the suppression of one of these hypothesis. This paper prsents a Markovian model under random environment called the Double Chain Markov Model which takes into account the maijn features of both visible and hidden models. Its main purpose is the modeling of non-homogeneous time-series. It is very flexible and can be estimated with traditional methods. The model is applied on a sequence of wind speeds and it appears to model data more successfully than both the usual Markov chains and hidden Markov models.     

Instability, Sensitivity, and Degeneracy of Discrete Exponential Families

In applications to dependent data, first and foremost relational data, a number of discrete exponential family models has turned out to be near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We introduce the notion of instability with an eye to characterize, detect, and penalize discrete exponential family models that are near-degenerate and problematic in terms of Markov chain Monte Carlo simulation and statistical inference. We show that unstable discrete exponential family models are characterized by excessive sensitivity and near-degeneracy. In special cases, the subset of the natural parameter space corresponding to non-degenerate distributions and mean-value parameters far from the boundary of the mean-value parameter space turns out to be a lower-dimensional subspace of the natural parameter space. These characteristics of unstable discrete exponential family models tend to obstruct Markov chain Monte Carlo simulation and statistical inference. In applications to relational data, we show that discrete exponential family models with Markov dependence tend to be unstable and that the parameter space of some curved exponential families contains unstable subsets.     

Non-mixture cure correlated frailty models in Bayesian approach

In this article, we develop a Bayesian approach for the estimation of two cure correlated frailty models that have been extended to the cure frailty models introduced by Yin [34]. We used the two different type of frailty with bivariate log-normal distribution instead of gamma distribution. A likelihood function was constructed based on a piecewise exponential distribution function. The model parameters were estimated by the Markov chain Monte Carlo method. The comparison of models is based on the Cox correlated frailty model with log-normal distribution. A real data set of bilateral corneal graft rejection was used to compare these models. The results of this data, based on deviance information criteria, showed the advantage of the proposed models.     

Inference from the Exponentiated Weibull Model with Applications to Real Data

In this paper, a novel Bayesian framework is used to derive the posterior density function, predictive density for a single future response, a bivariate future response, and several future responses from the exponentiated Weibull model (EWM). We study three related types of models, the exponentiated exponential, exponentiated Weibull, and beta generalized exponential, which are all utilized to determine the goodness of fit of two real data sets. The statistical analysis indicates that the EWM best fits both data sets. We determine the predictive means, standard deviations, highest predictive density intervals, and the shape characteristics for a single future response. We also consider a new parameterization method to determine the posterior kernel densities for the parameters. The summary results of the parameters are calculated by using the Markov chain Monte Carlo method.     

On additive and multiplicative damage models and the characterizations of linear and logarithmic exponential families

Rao (1963) has formulated a damage model which we call an additive damage model. A suitable damage model, which we call a multiplicative damage model, has been considered by Krishnaji (1970) for income-related problems. In these models, an original observation is subjected to damage, e.g., death or under-reporting, according to a specified probability law. Within the framework of an additive damage model, with a special form of damage, characterizations of the linear and logarithmic exponential families are formulated using regression properties of the damaged part on the undamaged part. The characterizations of the gamma and Pareto distributions that have been found of some use in the theory of income distributions, are obtained as special cases. Similar results are investigated within the framework of the multiplicative damage model.     

Analysing exponential family based messy data

Recently exponential family based random effects models have received considerable attention. These models usually arise from an unobservable random process added to the independent exponential family models. An unobservable correlated process, however, would cause correlations among the exponential family based data. This paper, first, develops an asymptotically optimal test for testing the appropriateness of a fixed effects model for the exponential family based independent data versus a random effects model for the exponential family based independent or correlated data. The paper, then, provides a general framework on regression analysis for the exponential family based data generated under the random effects models.     

Bayesian Inference for Contact Networks Given Epidemic Data

Abstract. In this article, we estimate the parameters of a simple random network and a stochastic epidemic on that network using data consisting of recovery times of infected hosts. The SEIR epidemic model we fit has exponentially distributed transmission times with Gamma distributed exposed and infectious periods on a network where every edge exists with the same probability, independent of other edges. We employ a Bayesian framework and Markov chain Monte Carlo (MCMC) integration to make estimates of the joint posterior distribution of the model parameters. We discuss the accuracy of the parameter estimates under various prior assumptions and show that it is possible in many scientifically interesting cases to accurately recover the parameters. We demonstrate our approach by studying a measles outbreak in Hagelloch, Germany, in 1861 consisting of 188 affected individuals. We provide an R package to carry out these analyses, which is available publicly on the Comprehensive R Archive Network.     

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.