Bayesian model selection for exponential random graph models via adjusted pseudolikelihoods

Models with intractable likelihood functions arise in areas including network analysis and spatial statistics, especially those involving Gibbs random fields. Posterior parameter es timation 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-normalised 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 paper specifies a method to adjust pseudolikelihoods in order 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.Comment: Supplementary material attached. To view attachments, please download and extract the gzzipped source file listed under "Other formats

Computationally efficient inference for latent position network models

Latent position models are widely used for the analysis of networks in a variety of research fields. In fact, these models possess a number of desirable theoretical properties, and are particularly easy to interpret. However, statistical methodologies to fit these models generally incur a computational cost which grows with the square of the number of nodes in the graph. This makes the analysis of large social networks impractical. In this paper, we propose a new method characterised by a linear computational complexity, which can be used to fit latent position models on networks of several tens of thousands nodes. Our approach relies on an approximation of the likelihood function, where the amount of noise introduced by the approximation can be arbitrarily reduced at the expense of computational efficiency. We establish several theoretical results that show how the likelihood error propagates to the invariant distribution of the Markov chain Monte Carlo sampler. In particular, we demonstrate that one can achieve a substantial reduction in computing time and still obtain a good estimate of the latent structure. Finally, we propose applications of our method to simulated networks and to a large coauthorships network, highlighting the usefulness of our approach.Comment: 39 pages, 10 figures, 1 tabl

Adaptive Incremental Mixture Markov chain Monte Carlo

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 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

Progressive hearing loss in Fabry's disease: a case report

Fabry's disease is a chromosomal X-linked inherited disease, which causes a lack of the lysosomal alpha-galactosidase A enzyme leading to a cellular accumulation of glycosphingolipids. This accumulation leads to various clinical disorders, including inner ear lesions, with sensorineural hearing loss and dizziness. This article proposes to describe a clinical case of a patient suffering from Fabry's disease with inner ear associated problems and to review the literature focusing on this subjec