Gareth Roberts

Gareth Roberts
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Gareth Roberts

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Pub Categories

Statistics - Computation (22)
Mathematics - Probability (20)
Statistics - Methodology (16)
Statistics - Theory (10)
Mathematics - Statistics (10)
Mathematics - Dynamical Systems (3)
Physics - Fluid Dynamics (2)
Mathematics - Classical Analysis and ODEs (2)
Quantitative Biology - Populations and Evolution (2)
Mathematics - Numerical Analysis (2)
Physics - Data Analysis; Statistics and Probability (1)
Physics - Physics and Society (1)
Physics - Atmospheric and Oceanic Physics (1)
Mathematical Physics (1)
Statistics - Machine Learning (1)
Mathematics - Algebraic Geometry (1)
Mathematics - Mathematical Physics (1)
Statistics - Applications (1)
Physics - Optics (1)
Instrumentation and Methods for Astrophysics (1)

Publications Authored By Gareth Roberts

We study collinear relative equilibria of the planar four-vortex problem where three of the four vortex strengths are identical. The $S_3$ invariance obtained from the equality of vorticities is used to reduce the defining equations and organize the solutions into two distinct groups based on the ordering of the vortices along the line. The number and type of solutions are given, along with a discussion of the bifurcations that occur. Read More

We study the convergence properties of the Gibbs Sampler in the context of posterior distributions arising from Bayesian analysis of Gaussian hierarchical models. We consider centred and non-centred parameterizations as well as their hybrids including the full family of partially non-centred parameterizations. We develop a novel methodology based on multi-grid decompositions to derive analytic expressions for the convergence rates of the algorithm for an arbitrary number of layers in the hierarchy, while previous work was typically limited to the two-level case. Read More

Piecewise deterministic Monte Carlo methods (PDMC) consist of a class of continuous-time Markov chain Monte Carlo methods (MCMC) which have recently been shown to hold considerable promise. Being non-reversible, the mixing properties of PDMC methods often significantly outperform classical reversible MCMC competitors. Moreover, in a Bayesian context they can use sub-sampling ideas, so that they need only access one data point per iteration, whilst still maintaining the true posterior distribution as their invariant distribution. Read More

Quasi-stationary distributions (QSDs)arise from stochastic processes that exhibit transient equilibrium behaviour on the way to absorption QSDs are often mathematically intractable and even drawing samples from them is not straightforward. In this paper the framework of Sequential Monte Carlo samplers is utilized to simulate QSDs and several novel resampling techniques are proposed to accommodate models with reducible state spaces, with particular focus on preserving particle diversity on discrete spaces. Finally an approach is considered to estimate eigenvalues associated with QSDs, such as the decay parameter. Read More

Recently there have been exciting developments in Monte Carlo methods, with the development of new MCMC and sequential Monte Carlo (SMC) algorithms which are based on continuous-time, rather than discrete-time, Markov processes. This has led to some fundamentally new Monte Carlo algorithms which can be used to sample from, say, a posterior distribution. Interestingly, continuous-time algorithms seem particularly well suited to Bayesian analysis in big-data settings as they need only access a small sub-set of data points at each iteration, and yet are still guaranteed to target the true posterior distribution. Read More

We provide quantitative bounds on the convergence to stationarity of real-valued Langevin diffusions with symmetric target densities. Read More

We prove that any four-body convex central configuration with perpendicular diagonals must be a kite configuration. The result extends to general power-law potential functions, including the planar four-vortex problem. Read More

This paper introduces a class of Monte Carlo algorithms which are based upon simulating a Markov process whose quasi-stationary distribution coincides with the distribution of interest. This differs fundamentally from, say, current Markov chain Monte Carlo in which we simulate a Markov chain whose stationary distribution is the target. We show how to approximate distributions of interest by carefully combining sequential Monte Carlo methods with methodology for the exact simulation of diffusions. Read More

A prompt public health response to a new epidemic relies on the ability to monitor and predict its evolution in real time as data accumulate. The 2009 A/H1N1 outbreak in the UK revealed pandemic data as noisy, contaminated and potentially biased, and originating from multiple sources, seriously questioning the capacity for real-time monitoring. Here we assess the feasibility of real-time inference based on such data by constructing an analytic tool combining an age-stratified SEIR transmission model with various observation models describing the data generation mechanisms. Read More

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multi-dimensional version of the Zig-Zag process of (Bierkens, Roberts, 2016), a continuous time piecewise deterministic Markov process. Read More

We present a description of our method to process a set of autocollimator-based deflectometer 1-dimensional line-scans taken over a large optical surface and reconstruct them to a best-fit conic-section surface. The challenge with our task is that each line-scan is in a different (unknown) coordinate reference frame with respect to the other line-scans in the set. This problem arises due to the limited angular measurement range of the autocollimator used in the deflectometer and the need to measure over a greater range; this results in the optic under measurement being rotated (in pitch and roll) between each scan to bring the autocollimator back into measurement range and therefore each scan is taken in a different coordinate frame. Read More

This paper considers the optimal scaling problem for high-dimensional random walk Metropolis algorithms for densities which are differentiable in Lp mean but which may be irregular at some points (like the Laplace density for example) and/or are supported on an interval. Our main result is the weak convergence of the Markov chain (appropriately rescaled in time and space) to a Langevin diffusion process as the dimension d goes to infinity. Because the log-density might be non-differentiable, the limiting diffusion could be singular. Read More

We introduce exact methods for the simulation of sample paths of one-dimensional diffusions with a discontinuity in the drift function. Our procedures require the simulation of finite-dimensional candidate draws from probability laws related to those of Brownian motion and its local time and are based on the principle of retrospective rejection sampling. A simple illustration is provided. Read More

In this paper we study the asymptotic behavior of the Random-Walk Metropolis algorithm on probability densities with two different `scales', where most of the probability mass is distributed along certain key directions with the `orthogonal' directions containing relatively less mass. Such class of probability measures arise in various applied contexts including Bayesian inverse problems where the posterior measure concentrates on a sub-manifold when the noise variance goes to zero. When the target measure concentrates on a linear sub-manifold, we derive analytically a diffusion limit for the Random-Walk Metropolis Markov chain as the scale parameter goes to zero. Read More

In Turitsyn, Chertkov, Vucelja (2011) a non-reversible Markov Chain Monte Carlo (MCMC) method on an augmented state space was introduced, here referred to as Lifted Metropolis-Hastings (LMH). A scaling limit of the magnetization process in the Curie-Weiss model is derived for LMH, as well as for Metropolis-Hastings (MH). The required jump rate in the high (supercritical) temperature regime equals $n^{1/2}$ for LMH, which should be compared to $n$ for MH. Read More

We introduce new Gaussian proposals to improve the efficiency of the standard Hastings-Metropolis algorithm in Markov chain Monte Carlo (MCMC) methods, used for the sampling from a target distribution in large dimension $d$. The improved complexity is $\mathcal{O}(d^{1/5})$ compared to the complexity $\mathcal{O}(d^{1/3})$ of the standard approach. We prove an asymptotic diffusion limit theorem and show that the relative efficiency of the algorithm can be characterised by its overall acceptance rate (with asymptotical value 0. Read More

Pseudo-marginal Markov chain Monte Carlo methods for sampling from intractable distributions have gained recent interest and have been theoretically studied in considerable depth. Their main appeal is that they are exact, in the sense that they target marginally the correct invariant distribution. However, the pseudo-marginal Markov chain can exhibit poor mixing and slow convergence towards its target. Read More

We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^2$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Read More

We connect known results about diffusion limits of Markov chain Monte Carlo (MCMC) algorithms to the Computer Science notion of algorithm complexity. Our main result states that any diffusion limit of a Markov process implies a corresponding complexity bound (in an appropriate metric). We then combine this result with previously-known MCMC diffusion limit results to prove that under appropriate assumptions, the Random-Walk Metropolis (RWM) algorithm in $d$ dimensions takes $O(d)$ iterations to converge to stationarity, while the Metropolis-Adjusted Langevin Algorithm (MALA) takes $O(d^{1/3})$ iterations to converge to stationarity. Read More

We consider whether ergodic Markov chains with bounded step size remain bounded in probability when their transitions are modified by an adversary on a bounded subset. We provide counterexamples to show that the answer is no in general, and prove theorems to show that the answer is yes under various additional assumptions. We then use our results to prove convergence of various adaptive Markov chain Monte Carlo algorithms. Read More

We derive new results comparing the asymptotic variance of diffusions by writing them as appropriate limits of discrete-time birth-death chains which themselves satisfy Peskun orderings. We then apply our results to simulated tempering algorithms to establish which choice of inverse temperatures minimises the asymptotic variance of all functionals and thus leads to the most efficient MCMC algorithm. Read More

A systematic Bayesian framework is developed for physics constrained parameter inference ofstochastic differential equations (SDE) from partial observations. The physical constraints arederived for stochastic climate models but are applicable for many fluid systems. A condition isderived for global stability of stochastic climate models based on energy conservation. Read More

We examine the behaviour of the pseudo-marginal random walk Metropolis algorithm, where evaluations of the target density for the accept/reject probability are estimated rather than computed precisely. Under relatively general conditions on the target distribution, we obtain limiting formulae for the acceptance rate and for the expected squared jump distance, as the dimension of the target approaches infinity, under the assumption that the noise in the estimate of the log-target is additive and is independent of the position. For targets with independent and identically distributed components, we also obtain a limiting diffusion for the first component. Read More

This paper introduces a framework for simulating finite dimensional representations of (jump) diffusion sample paths over finite intervals, without discretisation error (exactly), in such a way that the sample path can be restored at any finite collection of time points. Within this framework we extend existing exact algorithms and introduce novel adaptive approaches. We consider an application of the methodology developed within this paper which allows the simulation of upper and lower bounding processes which almost surely constrain (jump) diffusion sample paths to any specified tolerance. Read More

We study the linear and nonlinear stability of relative equilibria in the planar N-vortex problem, adapting the approach of Moeckel from the corresponding problem in celestial mechanics. After establishing some general theory, a topological approach is taken to show that for the case of positive circulations, a relative equilibrium is linearly stable if and only if it is a nondegenerate minimum of the Hamiltonian restricted to a level surface of the angular impulse (moment of inertia). Using a criterion of Dirichlet's, this implies that any linearly stable relative equilibrium with positive vorticities is also nonlinearly stable. Read More

We consider the optimal scaling problem for high-dimensional random walk Metropolis (RWM) algorithms where the target distribution has a discontinuous probability density function. Almost all previous analysis has focused upon continuous target densities. The main result is a weak convergence result as the dimensionality d of the target densities converges to infinity. Read More

We examine in detail the relative equilibria in the four-vortex problem where two pairs of vortices have equal strength, that is, \Gamma_1 = \Gamma_2 = 1 and \Gamma_3 = \Gamma_4 = m where m is a nonzero real parameter. One main result is that for m > 0, the convex configurations all contain a line of symmetry, forming a rhombus or an isosceles trapezoid. The rhombus solutions exist for all m but the isosceles trapezoid case exists only when m is positive. Read More

Contact tracing data collected from disease outbreaks has received relatively little attention in the epidemic modelling literature because it is thought to be unreliable: infection sources might be wrongly attributed, or data might be missing due to resource contraints in the questionnaire exercise. Nevertheless, these data might provide a rich source of information on disease transmission rate. This paper presents novel methodology for combining contact tracing data with rate-based contact network data to improve posterior precision, and therefore predictive accuracy. Read More

We present an iterative sampling method which delivers upper and lower bounding processes for the Brownian path. We develop such processes with particular emphasis on being able to unbiasedly simulate them on a personal computer. The dominating processes converge almost surely in the supremum and $L_1$ norms. Read More

We develop exact Markov chain Monte Carlo methods for discretely-sampled, directly and indirectly observed diffusions. The qualification "exact" refers to the fact that the invariant and limiting distribution of the Markov chains is the posterior distribution of the parameters free of any discretisation error. The class of processes to which our methods directly apply are those which can be simulated using the most general to date exact simulation algorithm. Read More

For a Markov transition kernel $P$ and a probability distribution $ \mu$ on nonnegative integers, a time-sampled Markov chain evolves according to the transition kernel $P_{\mu} = \sum_k \mu(k)P^k.$ In this note we obtain CLT conditions for time-sampled Markov chains and derive a spectral formula for the asymptotic variance. Using these results we compare efficiency of Barker's and Metropolis algorithms in terms of asymptotic variance. Read More

We consider various versions of adaptive Gibbs and Metropolis-within-Gibbs samplers, which update their selection probabilities (and perhaps also their proposal distributions) on the fly during a run by learning as they go in an attempt to optimize the algorithm. We present a cautionary example of how even a simple-seeming adaptive Gibbs sampler may fail to converge. We then present various positive results guaranteeing convergence of adaptive Gibbs samplers under certain conditions. Read More

The random walk Metropolis (RWM) is one of the most common Markov chain Monte Carlo algorithms in practical use today. Its theoretical properties have been extensively explored for certain classes of target, and a number of results with important practical implications have been derived. This article draws together a selection of new and existing key results and concepts and describes their implications. Read More

The science of networks has revolutionised research into the dynamics of interacting elements. It could be argued that epidemiology in particular has embraced the potential of network theory more than any other discipline. Here we review the growing body of research concerning the spread of infectious diseases on networks, focusing on the interplay between network theory and epidemiology. Read More

We consider Bayesian hierarchical models for survival analysis, where the survival times are modeled through an underlying diffusion process which determines the hazard rate. We show how these models can be efficiently treated by means of Markov chain Monte Carlo techniques. Read More

We investigate the properties of the Hybrid Monte-Carlo algorithm (HMC) in high dimensions. HMC develops a Markov chain reversible w.r. Read More

Scaling of proposals for Metropolis algorithms is an important practical problem in MCMC implementation. Criteria for scaling based on empirical acceptance rates of algorithms have been found to work consistently well across a broad range of problems. Essentially, proposal jump sizes are increased when acceptance rates are high and decreased when rates are low. Read More

We investigate local MCMC algorithms, namely the random-walk Metropolis and the Langevin algorithms, and identify the optimal choice of the local step-size as a function of the dimension $n$ of the state space, asymptotically as $n\to\infty$. We consider target distributions defined as a change of measure from a product law. Such structures arise, for instance, in inverse problems or Bayesian contexts when a product prior is combined with the likelihood. Read More

Assume that one aims to simulate an event of unknown probability $s\in (0,1)$ which is uniquely determined, however only its approximations can be obtained using a finite computational effort. Such settings are often encountered in statistical simulations. We consider two specific examples. Read More

We apply the analytic-numerical method of Roberts to determine the linear stability of time-reversible periodic simultaneous binary collision orbits in the symmetric collinear four body problem with masses 1, m, m, 1, and also in a symmetric planar four-body problem with equal masses. For the collinear problem, this verifies the earlier numerical results of Sweatman for linear stability. Read More

We introduce a powerful and flexible MCMC algorithm for stochastic simulation. The method builds on a pseudo-marginal method originally introduced in [Genetics 164 (2003) 1139--1160], showing how algorithms which are approximations to an idealized marginal algorithm, can share the same marginal stationary distribution as the idealized method. Theoretical results are given describing the convergence properties of the proposed method, and simple numerical examples are given to illustrate the promising empirical characteristics of the technique. Read More

This paper introduces a Monte Carlo method for maximum likelihood inference in the context of discretely observed diffusion processes. The method gives unbiased and a.s. Read More

We introduce a new property of Markov chains, called variance bounding. We prove that, for reversible chains at least, variance bounding is weaker than, but closely related to, geometric ergodicity. Furthermore, variance bounding is equivalent to the existence of usual central limit theorems for all $L^2$ functionals. Read More

We address the problem of parameter estimation for diffusion driven stochastic volatility models through Markov chain Monte Carlo (MCMC). To avoid degeneracy issues we introduce an innovative reparametrisation defined through transformations that operate on the time scale of the diffusion. A novel MCMC scheme which overcomes the inherent difficulties of time change transformations is also presented. Read More

We address the problem of likelihood based inference for correlated diffusion processes using Markov chain Monte Carlo (MCMC) techniques. Such a task presents two interesting problems. First, the construction of the MCMC scheme should ensure that the correlation coefficients are updated subject to the positive definite constraints of the diffusion matrix. Read More

Inference for Dirichlet process hierarchical models is typically performed using Markov chain Monte Carlo methods, which can be roughly categorised into marginal and conditional methods. The former integrate out analytically the infinite-dimensional component of the hierarchical model and sample from the marginal distribution of the remaining variables using the Gibbs sampler. Conditional methods impute the Dirichlet process and update it as a component of the Gibbs sampler. Read More

We characterise the convergence of the Gibbs sampler which samples from the joint posterior distribution of parameters and missing data in hierarchical linear models with arbitrary symmetric error distributions. We show that the convergence can be uniform, geometric or sub-geometric depending on the relative tail behaviour of the error distributions, and on the parametrisation chosen. Our theory is applied to characterise the convergence of the Gibbs sampler on latent Gaussian process models. Read More

In this paper we introduce a novel particle filter scheme for a class of partially-observed multivariate diffusions. %continuous-time dynamic models where the %signal is given by a multivariate diffusion process. We consider a variety of observation schemes, including diffusion observed with error, observation of a subset of the components of the multivariate diffusion and arrival times of a Poisson process whose intensity is a known function of the diffusion (Cox process). Read More

In this paper, we describe centering and noncentering methodology as complementary techniques for use in parametrization of broad classes of hierarchical models, with a view to the construction of effective MCMC algorithms for exploring posterior distributions from these models. We give a clear qualitative understanding as to when centering and noncentering work well, and introduce theory concerning the convergence time complexity of Gibbs samplers using centered and noncentered parametrizations. We give general recipes for the construction of noncentered parametrizations, including an auxiliary variable technique called the state-space expansion technique. Read More

A $\phi$-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace ``almost all'' by ``all,'' which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional Metropolis--Hastings algorithms are known to be Harris recurrent. Read More