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The relative abundance of groups of species is often used in ecological surveys to estimate community composition, a metric that reflects patterns of commonness and rarity of biological assemblages. The focus of this paper is measurements of the abundances of four benthic groups (that live on the seafloor) at several reefs on Australia’s Great Barrier Reef (GBR) gathered between 2012 and 2017. In this paper we develop a statistical model to find clusters of locations with similar composition. We examine the changes in clusters during a period impacted by an unprecedented sequence of extreme environmental disturbances. To achieve this, we propose a model that incorporates the geographical location of the data, accounting for the possibility that nearby reefs are similar in composition. This is accomplished with a Dirichlet mixture model and a Potts distribution on the cluster assignments. Non-availability of the normalised Potts distribution makes Bayesian inference a doubly-intractable task. To circumvent this additional inferential challenge, an approximate exchange algorithm is specified. The analysis of the 2012 data, collected before the weather disturbances, reveals four clusters. The four groups highlight the primary habitat patterns in the 2012 GBR, each with distinct ecological characteristics: (1) areas with above-average soft coral abundance, (2) sand-dominated regions commonly found in the central part, (3) southern reefs with a more balanced distribution of species, and (4) habitats dominated by algae and hard corals. Compared to subsequent surveys conducted after disturbances, there is evidence of a decline in the number of clusters and a simplification of reef composition at the regional scale.

We consider the task of simultaneous clustering of the two node sets involved in a bipartite network. The approach we adopt is based on use of the exact integrated complete likelihood for the latent blockmodel. Using this allows one to infer the number of clusters as well as cluster memberships using a greedy search. This gives a model-based clustering of the node sets. Experiments on simulated bipartite network data show that the greedy search approach is vastly more scalable than competing Markov chain Monte Carlo-based methods. Application to a number of real observed bipartite networks demonstrate the algorithms discussed.

Gibbs random fields play an important role in statistics. However, they are complicated to work with due to an intractability of the likelihood function and there has been much work devoted to finding computational algorithms to allow Bayesian inference to be conducted for such so-called doubly intractable distributions. This article extends this work and addresses the issue of estimating the evidence and Bayes factor for such models. The approach that we develop is shown to yield good performance. Supplementary materials for this article are available online.

Models with intractable likelihood functions arise in areas including network analysis and spatial statistics, especially those involving Gibbs random fields. Posterior parameter estimation in these settings is termed a doubly intractable problem because both the likelihood function and the posterior distribution are intractable. The comparison of Bayesian models is often based on the statistical evidence, the integral of the un-normalized posterior distribution over the model parameters which is rarely available in closed form. For doubly intractable models, estimating the evidence adds another layer of difficulty. Consequently, the selection of the model that best describes an observed network among a collection of exponential random graph models for network analysis is a daunting task. Pseudolikelihoods offer a tractable approximation to the likelihood but should be treated with caution because they can lead to an unreasonable inference. This article specifies a method to adjust pseudolikelihoods to obtain a reasonable, yet tractable, approximation to the likelihood. This allows implementation of widely used computational methods for evidence estimation and pursuit of Bayesian model selection of exponential random graph models for the analysis of social networks. Empirical comparisons to existing methods show that our procedure yields similar evidence estimates, but at a lower computational cost. Supplementary material for this article is available online.

We analyze the temporal bipartite network of the leading Irish companies and their directors from 2003 to 2013, encompassing the end of the Celtic Tiger boom and the ensuing financial crisis in 2008. We focus on the evolution of company interlocks, whereby a company director simultaneously sits on two ormore boards. We develop a statistical model for this dataset by embedding the positions of companies and directors in a latent space. The temporal evolution of the network is modeled through three levels of Markovian dependence: one on the model parameters, one on the companies’ latent positions, and one on the edges themselves. The model is estimated using Bayesian inference. Our analysis reveals that the level of interlocking, as measured by a contraction of the latent space, increased before and during the crisis, reaching a peak in 2009, and has generally stabilized since then.

Hamiltonian Monte Carlo (HMC) has been progressively incorporated within the statistician's toolbox as an alternative sampling method in settings when standard Metropolis-Hastings is inefficient. HMC generates a Markov chain on an augmented state space with transitions based on a deterministic differential flow derived from Hamiltonian mechanics. In practice, the evolution of Hamiltonian systems cannot be solved analytically, requiring numerical integration schemes. Under numerical integration, the resulting approximate solution no longer preserves the measure of the target distribution, therefore an accept-reject step is used to correct the bias. For doubly intractable distributions-such as posterior distributions based on Gibbs random fields-HMC suffers from some computational difficulties: computation of gradients in the differential flow and computation of the accept-reject proposals poses difficulty. In this article, we study the behavior of HMC when these quantities are replaced by Monte Carlo estimates. Supplemental codes for implementing methods used in the article are available online.

We derive properties of latent variable models for networks, a broad class of models that includes the widely used latent position models. We characterize several features of interest, with particular focus on the degree distribution, clustering coefficient, average path length, and degree correlations. We introduce the Gaussian latent position model, and derive analytic expressions and asymptotic approximations for its network properties. We pay particular attention to one special case, the Gaussian latent position model with random effects, and show that it can represent the heavy-tailed degree distributions, positive asymptotic clustering coefficients, and small-world behaviors that often occur in observed social networks. Finally, we illustrate the ability of the models to capture important features of real networks through several well-known datasets.

Latent stochastic blockmodels are flexible statistical models that are widely used in social network analysis. In recent years, efforts have been made to extend these models to temporal dynamic networks, whereby the connections between nodes are observed at a number of different times. In this paper, we propose a new Bayesian framework to characterize the construction of connections. We rely on a Markovian property to describe the evolution of nodes' cluster memberships over time. We recast the problem of clustering the nodes of the network into a model-based context, showing that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximize this quantity, thereby estimating both the optimal partition and the ideal number of groups in a single inferential framework. Finally, we propose applications of our methodology to both real and artificial datasets.

We propose adaptive incremental mixture Markov chain Monte Carlo (AIMM), a novel approach to sample from challenging probability distributions defined on a general state-space. While adaptive MCMC methods usually update a parametric proposal kernel with a global rule, AIMM locally adapts a semiparametric kernel. AIMM is based on an independent Metropolis-Hastings proposal distribution which takes the form of a finite mixture of Gaussian distributions. Central to this approach is the idea that the proposal distribution adapts to the target by locally adding a mixture component when the discrepancy between the proposal mixture and the target is deemed to be too large. As a result, the number of components in the mixture proposal is not fixed in advance. Theoretically, we prove that there exists a stochastic process that can be made arbitrarily close to AIMM and that converges to the correct target distribution. We also illustrate that it performs well in practice in a variety of challenging situations, including high-dimensional and multimodal target distributions. Finally, the methodology is successfully applied to two real data examples, including the Bayesian inference of a semiparametric regression model for the Boston Housing dataset. Supplementary materials for this article are available online.