arXiv subjects

Nial Friel

Publications and source records attributed to Nial Friel.

At least 37 records · Page 2Linked to original sources

arXiv2020

Bayesian inference, model selection and likelihood estimation using fast rejection sampling: the Conway-Maxwell-Poisson distribution

Bayesian inference for models with intractable likelihood functions represents a challenging suite of problems in modern statistics. In this work we analyse the Conway-Maxwell-Poisson (COM-Poisson) distribution, a two parameter generalisation of the Poisson distribution. COM-Poisson regression modelling allows the flexibility to model dispersed count data as part of a generalised linear model (GLM) with a COM-Poisson response, where exogenous covariates control the mean and dispersion level of the response. The major difficulty with COM-Poisson regression is that the likelihood function contains multiple intractable normalising constants and is not amenable to standard inference and MCMC techniques. Recent work by Chanialidis et al. (2017) has seen the development of a sampler to draw random variates from the COM-Poisson likelihood using a rejection sampling algorithm. We provide a new rejection sampler for the COM-Poisson distribution which significantly reduces the CPU time required to perform inference for COM-Poisson regression models. A novel extension of this work shows that for any intractable likelihood function with an associated rejection sampler it is possible to construct unbiased estimators of the intractable likelihood which proves useful for model selection or for use within pseudo-marginal MCMC algorithms (Andrieu and Roberts, 2009). We demonstrate all of these methods on a real-world dataset of takeover bids.

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arXiv2019

Bayesian variational inference for exponential random graph models

Deriving Bayesian inference for exponential random graph models (ERGMs) is a challenging "doubly intractable" problem as the normalizing constants of the likelihood and posterior density are both intractable. Markov chain Monte Carlo (MCMC) methods which yield Bayesian inference for ERGMs, such as the exchange algorithm, are asymptotically exact but computationally intensive, as a network has to be drawn from the likelihood at every step using, for instance, a "tie no tie" sampler. In this article, we develop a variety of variational methods for Gaussian approximation of the posterior density and model selection. These include nonconjugate variational message passing based on an adjusted pseudolikelihood and stochastic variational inference. To overcome the computational hurdle of drawing a network from the likelihood at each iteration, we propose stochastic gradient ascent with biased but consistent gradient estimates computed using adaptive self-normalized importance sampling. These methods provide attractive fast alternatives to MCMC for posterior approximation. We illustrate the variational methods using real networks and compare their accuracy with results obtained via MCMC and Laplace approximation.

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arXiv2018

Model comparison for Gibbs random fields using noisy reversible jump Markov chain Monte Carlo

The reversible jump Markov chain Monte Carlo (RJMCMC) method offers an across-model simulation approach for Bayesian estimation and model comparison, by exploring the sampling space that consists of several models of possibly varying dimensions. A naive implementation of RJMCMC to models like Gibbs random fields suffers from computational difficulties: the posterior distribution for each model is termed doubly-intractable since computation of the likelihood function is rarely available. Consequently, it is simply impossible to simulate a transition of the Markov chain in the presence of likelihood intractability. A variant of RJMCMC is presented, called noisy RJMCMC, where the underlying transition kernel is replaced with an approximation based on unbiased estimators. Based on previous theoretical developments, convergence guarantees for the noisy RJMCMC algorithm are provided. The experiments show that the noisy RJMCMC algorithm can be much more efficient than other exact methods, provided that an estimator with controlled Monte Carlo variance is used, a fact which is in agreement with the theoretical analysis.

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arXiv2018

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.

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arXiv2018

Informed Sub-Sampling MCMC: Approximate Bayesian Inference for Large Datasets

This paper introduces a framework for speeding up Bayesian inference conducted in presence of large datasets. We design a Markov chain whose transition kernel uses an (unknown) fraction of (fixed size) of the available data that is randomly refreshed throughout the algorithm. Inspired by the Approximate Bayesian Computation (ABC) literature, the subsampling process is guided by the fidelity to the observed data, as measured by summary statistics. The resulting algorithm, Informed Sub-Sampling MCMC (ISS-MCMC), is a generic and flexible approach which, contrary to existing scalable methodologies, preserves the simplicity of the Metropolis-Hastings algorithm. Even though exactness is lost, i.e. the chain distribution approximates the posterior, we study and quantify theoretically this bias and show on a diverse set of examples that it yields excellent performances when the computational budget is limited. If available and cheap to compute, we show that setting the summary statistics as the maximum likelihood estimator is supported by theoretical arguments.

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arXiv2018

Discussion of the article "Bayesian cluster analysis: point estimation and credible balls" by Wade and Ghahramani

We present a discussion of the paper "Bayesian cluster analysis: point estimation and credible balls" by Wade and Ghahramani. We believe that this paper contributes substantially to the literature on Bayesian clustering by filling in an important methodological gap, by providing a means to assess the uncertainty around a point estimate of the optimal clustering solution based on a given loss function. In our discussion we reflect on the characterisation of uncertainty around the Bayesian optimal partition, revealing other possible alternatives that may be viable. In addition, we suggest other important extensions of the approach proposed which may lead to wider applicability.

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arXiv2018

Noisy Hamiltonian Monte Carlo for doubly-intractable distributions

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 paper, we study the behaviour of HMC when these quantities are replaced by Monte Carlo estimates.

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arXiv2017

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.

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arXiv2017

Efficient MCMC for Gibbs Random Fields using pre-computation

Bayesian inference of Gibbs random fields (GRFs) is often referred to as a doubly intractable problem, since the likelihood function is intractable. The exploration of the posterior distribution of such models is typically carried out with a sophisticated Markov chain Monte Carlo (MCMC) method, the exchange algorithm (Murray et al., 2006), which requires simulations from the likelihood function at each iteration. The purpose of this paper is to consider an approach to dramatically reduce this computational overhead. To this end we introduce a novel class of algorithms which use realizations of the GRF model, simulated offline, at locations specified by a grid that spans the parameter space. This strategy speeds up dramatically the posterior inference, as illustrated on several examples. However, using the pre-computed graphs introduces a noise in the MCMC algorithm, which is no longer exact. We study the theoretical behaviour of the resulting approximate MCMC algorithm and derive convergence bounds using a recent theoretical development on approximate MCMC methods.

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arXiv2017

Efficient Bayesian inference for exponential random graph models by correcting the pseudo-posterior distribution

Exponential random graph models are an important tool in the statistical analysis of data. However, Bayesian parameter estimation for these models is extremely challenging, since evaluation of the posterior distribution typically involves the calculation of an intractable normalizing constant. This barrier motivates the consideration of tractable approximations to the likelihood function, such as the pseudolikelihood function, which offers an approach to constructing such an approximation. Naive implementation of what we term a pseudo-posterior resulting from replacing the likelihood function in the posterior distribution by the pseudolikelihood is likely to give misleading inferences. We provide practical guidelines to correct a sample from such a pseudo-posterior distribution so that it is approximately distributed from the target posterior distribution and discuss the computational and statistical efficiency that result from this approach. We illustrate our methodology through the analysis of real-world graphs. Comparisons against the approximate exchange algorithm of Caimo and Friel (2011) are provided, followed by concluding remarks.

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arXiv2017

Bergm: Bayesian exponential random graph models in R

The Bergm package provides a comprehensive framework for Bayesian inference using Markov chain Monte Carlo (MCMC) algorithms. It can also supply graphical Bayesian goodness-of-fit procedures that address the issue of model adequacy. The package is simple to use and represents an attractive way of analysing network data as it offers the advantage of a complete probabilistic treatment of uncertainty. Bergm is based on the ergm package and therefore it makes use of the same model set-up and network simulation algorithms. The Bergm package has been continually improved in terms of speed performance over the last years and now offers the end-user a feasible option for carrying out Bayesian inference for networks with several thousands of nodes.

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arXiv2017

Optimal Bayesian estimators for latent variable cluster models

In cluster analysis interest lies in probabilistically capturing partitions of individuals, items or observations into groups, such that those belonging to the same group share similar attributes or relational profiles. Bayesian posterior samples for the latent allocation variables can be effectively obtained in a wide range of clustering models, including finite mixtures, infinite mixtures, hidden Markov models and block models for networks. However, due to the categorical nature of the clustering variables and the lack of scalable algorithms, summary tools that can interpret such samples are not available. We adopt a Bayesian decision theoretic approach to define an optimality criterion for clusterings, and propose a fast and context-independent greedy algorithm to find the best allocations. One important facet of our approach is that the optimal number of groups is automatically selected, thereby solving the clustering and the model-choice problems at the same time. We consider several loss functions to compare partitions, and show that our approach can accommodate a wide range of cases. Finally, we illustrate our approach on a variety of real-data applications for three different clustering models: Gaussian finite mixtures, stochastic block models and latent block models for networks.

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arXiv2017

Choosing the number of groups in a latent stochastic block model for dynamic networks

Latent stochastic block models 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 extend the original stochastic block model by using 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, and show that the integrated completed likelihood can be evaluated analytically for a number of likelihood models. Then, we propose a scalable greedy algorithm to maximise 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.

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arXiv2017

An adaptive MCMC method for multiple changepoint analysis with applications to large datasets

We consider the problem of Bayesian inference for changepoints where the number and position of the changepoints are both unknown. In particular, we consider product partition models where it is possible to integrate out model parameters for the regime between each changepoint, leaving a posterior distribution over a latent vector indicating the presence or not of a changepoint at each observation. The same problem setting has been considered by Fearnhead (2006) where one can use filtering recursions to make exact inference. However the complexity of this algorithm depends quadratically on the number of observations. Our approach relies on an adaptive Markov Chain Monte Carlo (MCMC) method for finite discrete state spaces. We develop an adaptive algorithm which can learn from the past states of the Markov chain in order to build proposal distributions which can quickly discover where changepoint are likely to be located. We prove that our algorithm leaves the posterior distribution ergodic. Crucially, we demonstrate that our adaptive MCMC algorithm is viable for large datasets for which the filtering recursions approach is not. Moreover, we show that inference is possible in a reasonable time.

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arXiv2017

Bayesian model selection for the latent position cluster model for Social Networks

The latent position cluster model is a popular model for the statistical analysis of network data. This model assumes that there is an underlying latent space in which the actors follow a finite mixture distribution. Moreover, actors which are close in this latent space are more likely to be tied by an edge. This is an appealing approach since it allows the model to cluster actors which consequently provides the practitioner with useful qualitative information. However, exploring the uncertainty in the number of underlying latent components in the mixture distribution is a complex task. The current state-of-the-art is to use an approximate form of BIC for this purpose, where an approximation of the log-likelihood is used instead of the true log-likelihood which is unavailable. The main contribution of this paper is to show that through the use of conjugate prior distributions it is possible to analytically integrate out almost all of the model parameters, leaving a posterior distribution which depends on the allocation vector of the mixture model. This enables posterior inference over the number of components in the latent mixture distribution without using trans- dimensional MCMC algorithms such as reversible jump MCMC. Our approach is compared with the state-of-the-art latentnet (Krivitsky & Handcock 2015) and VBLPCM (Salter-Townshend & Murphy 2013) packages.

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arXiv2015

Light and Widely Applicable MCMC: Approximate Bayesian Inference for Large Datasets

Light and Widely Applicable (LWA-) MCMC is a novel approximation of the Metropolis-Hastings kernel targeting a posterior distribution defined on a large number of observations. Inspired by Approximate Bayesian Computation, we design a Markov chain whose transition makes use of an unknown but fixed, fraction of the available data, where the random choice of sub-sample is guided by the fidelity of this sub-sample to the observed data, as measured by summary (or sufficient) statistics. LWA-MCMC is a generic and flexible approach, as illustrated by the diverse set of examples which we explore. In each case LWA-MCMC yields excellent performance and in some cases a dramatic improvement compared to existing methodologies.

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arXiv2015

Calibration of conditional composite likelihood for Bayesian inference on Gibbs random fields

Gibbs random fields play an important role in statistics, however, the resulting likelihood is typically unavailable due to an intractable normalizing constant. Composite likelihoods offer a principled means to construct useful approximations. This paper provides a mean to calibrate the posterior distribution resulting from using a composite likelihood and illustrate its performance in several examples.

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arXiv2015

Properties of Latent Variable Network Models

We derive properties of Latent Variable Models for networks, a broad class of models that includes the widely-used Latent Position Models. These include the average 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 Models with Random Effects, and show that it can represent the heavy-tailed degree distributions, positive asymptotic clustering coefficients and small-world behaviours that are often observed in social networks. Several real and simulated examples illustrate the ability of the models to capture important features of observed networks.

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