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Nonparametric Bayes Modeling of Populations of Networks
, 2014
"... Collections of networks are available in many research fields. In connectomic applications, interconnections among brain regions are collected from each patient, with interest focusing on studying shared structure and the population distribution of deviations across individuals. Current methods foc ..."
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Collections of networks are available in many research fields. In connectomic applications, interconnections among brain regions are collected from each patient, with interest focusing on studying shared structure and the population distribution of deviations across individuals. Current methods focus on reducing network data to features prior to statistical analysis, while we propose a fully generative Bayesian nonparametric approach for modeling the population distribution of networkvalued data. The joint distribution of the edges follows a multivariate Bernoulli distribution, with transformed edge probability vectors expressed as the sum of a shared similarity vector and a classspecific deviation modeled via flexible lowrank factorization exploiting the network structure. The formulation is provably flexible, leads to a simple and computationally efficient Gibbs sampler, and provides a framework for clustering graphvalued data, while inferring a clusterspecific rank. We discuss theoretical properties and illustrate the performance in simulations and application to human brain network data.
Markov chain Monte Carlo with linchpin variables
, 2014
"... Many posteriors can be factored into a product of a conditional density which is easy to sample directly and a marginal density. If it is possible to make a draw from the marginal, then a simple sequential sampling algorithm can be used to make a perfect draw from the joint target density. When the ..."
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Many posteriors can be factored into a product of a conditional density which is easy to sample directly and a marginal density. If it is possible to make a draw from the marginal, then a simple sequential sampling algorithm can be used to make a perfect draw from the joint target density. When the marginal is difficult to sample from we propose to use a MetropolisHastings step on the marginal followed by a draw from the conditional distribution. We show that the resulting Markov chain, called a linchpin variable sampler, is reversible and that its convergence rate is the same as that of the subchain where the MetropolisHastings step is being performed. We use this to construct uniformly ergodic linchpin variable samplers for two versions of a Bayesian linear mixed model and a Bayesian probit regression model.
Bayesian Logistic Gaussian Process Models for Dynamic Networks
"... Timevarying adjacency matrices encoding the presence or absence of a relation among entities are available in many research fields. Motivated by an application to studying dynamic networks among sports teams, we propose a Bayesian nonparametric model. The proposed approach uses a logistic mapping ..."
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Timevarying adjacency matrices encoding the presence or absence of a relation among entities are available in many research fields. Motivated by an application to studying dynamic networks among sports teams, we propose a Bayesian nonparametric model. The proposed approach uses a logistic mapping from the probability matrix, encoding link probabilities between each team, to an embedded latent relational space. Within this latent space, we incorporate a dictionary of Gaussian process (GP) latent trajectories characterizing changes over time in each team, while allowing learning of the number of latent dimensions through a specially tailored prior for the GP covariance. The model is provably flexible and borrows strength across the network and over time. We provide simulation experiments and an application to the Italian soccer Championship. 1
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"... Bayesian dynamic financial networks with timevarying predictors Bayesian dynamic financial networks with timevarying predictors ..."
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Bayesian dynamic financial networks with timevarying predictors Bayesian dynamic financial networks with timevarying predictors
Printed in Great Britain Nonparametric Bayes dynamic modelling of relational data
"... Symmetric binary matrices representing relations are collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being on inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionali ..."
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Symmetric binary matrices representing relations are collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being on inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lowerdimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the link probability matrix space to the latent relational space, we obtain a flexible and computationally tractable formulation. Employing Pólyagamma data augmentation, an efficient Gibbs sampler is developed for posterior computation, with the dimension of the latent space automatically inferred. We provide theoretical results on flexibility of the model, and illustrate its performance via simulation experiments. We also consider an application to comovements in world financial markets.
Printed in Great Britain Advance Access publication 8 October 2014 Nonparametric Bayes dynamic modelling of relational data
"... Symmetric binary matrices representing relations are collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being on inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionali ..."
Abstract
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Symmetric binary matrices representing relations are collected in many areas. Our focus is on dynamically evolving binary relational matrices, with interest being on inference on the relationship structure and prediction. We propose a nonparametric Bayesian dynamic model, which reduces dimensionality in characterizing the binary matrix through a lowerdimensional latent space representation, with the latent coordinates evolving in continuous time via Gaussian processes. By using a logistic mapping function from the link probability matrix space to the latent relational space, we obtain a flexible and computationally tractable formulation. Employing Pólyagamma data augmentation, an efficient Gibbs sampler is developed for posterior computation, with the dimension of the latent space automatically inferred. We provide theoretical results on flexibility of the model, and illustrate its performance via simulation experiments. We also consider an application to comovements in world financial markets.