Publications
35 results found
Rougier J, Duncan A, 2024, Bayesian modeling and inference for one-shot experiments, Technometrics, Vol: 66, Pages: 55-64, ISSN: 0040-1706
In one-shot experiments, units are subjected to varying levels of stimulus and their binary response (go/no-go) is recorded. Experimental data is used to estimate the “sensitivity function”, which characterizes the probability of a “go” for a given level of stimulus. We review the current GLM approaches to modeling and inference, and identify some deficiencies. To address these, we propose a novel Bayesian approach using an adjustable number of cubic splines, with physically-plausible smoothness, monotonicity, and tail constraints introduced through the prior distribution on the coefficients. Our approach runs “out of the box,” and in roughly the same time as the GLM approaches. We illustrate with two contrasting datasets, and show that our more flexible Bayesian approach gives different inferences to the GLM approaches for both the sensitivity function and its inverse.
Liu X, Duncan A, Gandy A, 2023, Using Perturbation to Improve Goodness-of-Fit Tests based on Kernelized Stein Discrepancy, Fortieth International Conference on Machine Learning
Bull LA, Di Francesco D, Dhada M, et al., 2023, Hierarchical Bayesian modeling for knowledge transfer across engineering fleets via multitask learning, Computer-Aided Civil and Infrastructure Engineering, Vol: 38, Pages: 821-848, ISSN: 1093-9687
A population-level analysis is proposed to address data sparsity when building predictive models for engineering infrastructure. Utilizing an interpretable hierarchical Bayesian approach and operational fleet data, domain expertise is naturally encoded (and appropriately shared) between different subgroups, representing (1) use-type, (2) component, or (3) operating condition. Specifically, domain expertise is exploited to constrain the model via assumptions (and prior distributions) allowing the methodology to automatically share information between similar assets, improving the survival analysis of a truck fleet (15% and 13% increases in predictive log-likelihood of hazard) and power prediction in a wind farm (up to 82% reduction in the standard deviation of maximum output prediction). In each asset management example, a set of correlated functions is learnt over the fleet, in a combined inference, to learn a population model. Parameter estimation is improved when subfleets are allowed to share correlated information at different levels in the hierarchy; the (averaged) reduction in standard deviation for interpretable parameters in the survival analysis is 70%, alongside 32% in wind farm power models. In turn, groups with incomplete data automatically borrow statistical strength from those that are data-rich. The statistical correlations enable knowledge transfer via Bayesian transfer learning, and the correlations can be inspected to inform which assets share information for which effect (i.e., parameter). Successes in both case studies demonstrate the wide applicability in practical infrastructure monitoring, since the approach is naturally adapted between interpretable fleet models of different in situ examples.
Duncan AB, Duong MH, Pavliotis GA, 2023, Brownian motion in an N-scale periodic potential, Journal of Statistical Physics, Vol: 190, Pages: 1-34, ISSN: 0022-4715
We study the problem of Brownian motion in a multiscale potential. The potential is assumed to have N+1 scales (i.e. N small scales and one macroscale) and to depend periodically on all the small scales. We show that for nonseparable potentials, i.e. potentials in which the microscales and the macroscale are fully coupled, the homogenized equation is an overdamped Langevin equation with multiplicative noise driven by the free energy, for which the detailed balance condition still holds. This means, in particular, that homogenized dynamics is reversible and that the coarse-grained Fokker–Planck equation is still a Wasserstein gradient flow with respect to the coarse-grained free energy. The calculation of the effective diffusion tensor requires the solution of a system of N coupled Poisson equations.
Huang KH, Liu X, Duncan AB, et al., 2023, A High-dimensional Convergence Theorem for U-statistics with Applications to Kernel-based Testing, Pages: 3827-3918
We prove a convergence theorem for U-statistics of degree two, where the data dimension d is allowed to scale with sample size n. We find that the limiting distribution of a U-statistic undergoes a phase transition from the non-degenerate Gaussian limit to the degenerate limit, regardless of its degeneracy and depending only on a moment ratio. A surprising consequence is that a non-degenerate U-statistic in high dimensions can have a non-Gaussian limit with a larger variance and asymmetric distribution. Our bounds are valid for any finite n and d, independent of individual eigenvalues of the underlying function, and dimension-independent under a mild assumption. As an application, we apply our theory to two popular kernel-based distribution tests, MMD and KSD, whose high-dimensional performance has been challenging to study. In a simple empirical setting, our results correctly predict how the test power at a fixed threshold scales with d and the bandwidth.
Bull LA, Abdallah I, Mylonas C, et al., 2023, Data-Centric Monitoring of Wind Farms: Combining Sources of Information, Data Driven Methods for Civil Structural Health Monitoring and Resilience: Latest Developments and Applications, Pages: 120-180, ISBN: 9781032308371
This chapter summarizes recent developments in data-centric monitoring of wind farms. We present methodologies which share information from multiple sources. Problems include inter-turbine modeling of wind speed and wake effects; methods for combining multiple emulators; spatially distributed virtual sensing, for the estimation of dynamic loads; hierarchical Bayesian modeling of power relationships at the systems level. We demonstrate how an increasingly holistic approach to wind farm monitoring brings a series of advantages.
Di Francesco D, Girolami M, Duncan AB, et al., 2022, A probabilistic model for quantifying uncertainty in the failure assessment diagram, STRUCTURAL SAFETY, Vol: 99, ISSN: 0167-4730
Seshadri P, Duncan A, Thorne G, et al., 2022, Bayesian assessments of aeroengine performance with transfer learning, Data-Centric Engineering, Vol: 3, Pages: e29-1-e29-30, ISSN: 2632-6736
Aeroengine performance is determined by temperature and pressure profiles along various axial stations withinan engine. Given limited sensor measurements both along and between axial stations, we require a statisticallyprincipled approach for inferring these profiles. In this paper we detail a Bayesian methodology for interpolatingthe spatial temperature or pressure profile at axial stations within an aeroengine. The profile at any given axialstation is represented as a spatial Gaussian random field on an annulus, with circumferential variations modelledusing a Fourier basis and radial variations modelled with a squared exponential kernel. This Gaussian randomfield is extended to ingest data from multiple axial measurement planes, with the aim of transferring informationacross the planes. To facilitate this type of transfer learning, a novel planar covariance kernel is proposed, withhyperparameters that characterise the correlation between any two measurement planes. In the scenario where frequencies comprising the temperature field are unknown, we utilise a sparsity-promoting prior on the frequencies toencourage sparse representations. This easily extends to cases with multiple engine planes whilst accommodatingfrequency variations between the planes. The main quantity of interest, the spatial area average is readily obtainedin closed form. We term this the Bayesian area average and demonstrate how this metric offers far more representative averages than a sector area average—a widely used area averaging approach. Furthermore, the Bayesian areaaverage naturally decomposes the posterior uncertainty into terms characterising insufficient sampling and sensormeasurement error respectively. This too provides a significant improvement over prior standard deviation baseduncertainty breakdowns.
Seshadri P, Duncan A, Thorne G, 2022, Bayesian Mass Averaging in Rigs and Engines, JOURNAL OF TURBOMACHINERY-TRANSACTIONS OF THE ASME, Vol: 144, ISSN: 0889-504X
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- Citations: 2
Wong CY, Seshadri P, Scillitoe A, et al., 2022, Blade Envelopes Part I: Concept and Methodology, JOURNAL OF TURBOMACHINERY-TRANSACTIONS OF THE ASME, Vol: 144, ISSN: 0889-504X
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- Citations: 3
Wong CY, Seshadri P, Scillitoe A, et al., 2022, Blade Envelopes Part II: Multiple Objectives and Inverse Design, JOURNAL OF TURBOMACHINERY-TRANSACTIONS OF THE ASME, Vol: 144, ISSN: 0889-504X
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- Citations: 2
Yatsyshin P, Kalliadasis S, Duncan AB, 2022, Physics-constrained Bayesian inference of state functions in classical density-functional theory, Journal of Chemical Physics, Vol: 156, Pages: 074105-1-074105-10, ISSN: 0021-9606
We develop a novel data-driven approach to the inverse problem of classical statistical mechanics: given experimental data on the collective motion of a classical many-body system, how does one characterise the free energy landscape of that system? By combining non-parametric Bayesian inference with physically-motivated constraints, we develop an efficient learning algorithm which automates the construction of approximate free energy functionals. In contrast to optimisation-based machine learning approaches, which seek to minimise a cost function, the centralidea of the proposed Bayesian inference is to propagate a set of prior assumptions through the model, derived from physical principles. The experimental data is usedto probabilistically weigh the possible model predictions. This naturally leads to humanly interpretable algorithms with full uncertainty quantification of predictions. In our case, the output of the learning algorithm is a probability distribution over a family of free energy functionals, consistent with the observed particle data. We find that surprisingly small data samples contain sufficient information for inferring highly accurate analytic expressions of the underlying free energy functionals, making our algorithm highly data efficient. We consider excluded volume particle interactions, which are ubiquitous in nature, whilst being highly challenging for modelling in terms of free energy. To validate our approach we consider the paradigmaticcase of one-dimensional fluid and develop inference algorithms for the canonical and grand-canonical statistical-mechanical ensembles. Extensions to higher dimensional systems are conceptually straightforward, whilst standard coarse-graining techniques allow one to easily incorporate attractive interactions
Briol F-X, Barp A, Duncan AB, et al., 2022, Statistical inference for generative models with maximum meandiscrepancy, Publisher: ArXiv
While likelihood-based inference and its variants provide a statisticallyefficient and widely applicable approach to parametric inference, theirapplication to models involving intractable likelihoods poses challenges. Inthis work, we study a class of minimum distance estimators for intractablegenerative models, that is, statistical models for which the likelihood isintractable, but simulation is cheap. The distance considered, maximum meandiscrepancy (MMD), is defined through the embedding of probability measuresinto a reproducing kernel Hilbert space. We study the theoretical properties ofthese estimators, showing that they are consistent, asymptotically normal androbust to model misspecification. A main advantage of these estimators is theflexibility offered by the choice of kernel, which can be used to trade-offstatistical efficiency and robustness. On the algorithmic side, we study thegeometry induced by MMD on the parameter space and use this to introduce anovel natural gradient descent-like algorithm for efficient implementation ofthese estimators. We illustrate the relevance of our theoretical results onseveral classes of models including a discrete-time latent Markov process andtwo multivariate stochastic differential equation models.
Liu X, Zhu H, Ton J-F, et al., 2022, Grassmann stein variational gradient descent, International Conference on Artificial Intelligence and Statistics, Publisher: JMLR-JOURNAL MACHINE LEARNING RESEARCH, Pages: 1-20, ISSN: 2640-3498
Stein variational gradient descent (SVGD) is a deterministic particle inference algorithm that provides an efficient alternative to Markov chain Monte Carlo. However, SVGD has been found to suffer from variance underestimation when the dimensionality of the target distribution is high. Recent developments have advocated projecting both the score function and the data onto real lines to sidestep this issue, although this can severely overestimate the epistemic (model) uncertainty. In this work, we propose Grassmann Stein variational gradient descent (GSVGD) as an alternative approach, which permits projections onto arbitrary dimensional subspaces. Compared with other variants of SVGD that rely on dimensionality reduction, GSVGD updates the projectors simultaneously for the score function and the data, and the optimal projectors are determined through a coupled Grassmann-valued diffusion process which explores favourable subspaces. Both our theoretical and experimental results suggest that GSVGD enjoys efficient state-space exploration in high-dimensional problems that have an intrinsic low-dimensional structure.
Scillitoe A, Seshadri P, Wong CY, et al., 2021, Polynomial ridge flowfield estimation, PHYSICS OF FLUIDS, Vol: 33, ISSN: 1070-6631
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- Citations: 1
Cockayne J, Duncan A, 2021, Probabilistic gradients for fast calibration of differential equation models, SIAM/ASA Journal on Uncertainty Quantification, Vol: 9, ISSN: 2166-2525
Calibration of large-scale differential equation models to observational or experimental data is a widespread challenge throughout applied sciences and engineering. A crucial bottleneck in state-of-the art calibration methods is the calculation of local sensitivities, i.e. derivatives of the loss function with respect to the estimated parameters, which often necessitates several numerical solves of the underlying system of partial or ordinary differential equations. In this paper we present a new probabilistic approach to computing local sensitivities. The proposed method has several advantages over classical methods. Firstly, it operates within a constrained computational budget and provides a probabilistic quantification of uncertainty incurred in the sensitivities from this constraint. Secondly, information from previous sensitivity estimates can be recycled in subsequent computations, reducing the overall computational effort for iterative gradient-based calibration methods. The methodology presented is applied to two challenging test problems and compared against classical methods.
Wilde H, Mellan T, Hawryluk I, et al., 2021, The association between mechanical ventilator compatible bed occupancy and mortality risk in intensive care patients with COVID-19: a national retrospective cohort study., BMC Medicine, Vol: 19, Pages: 1-12, ISSN: 1741-7015
BACKGROUND: The literature paints a complex picture of the association between mortality risk and ICU strain. In this study, we sought to determine if there is an association between mortality risk in intensive care units (ICU) and occupancy of beds compatible with mechanical ventilation, as a proxy for strain. METHODS: A national retrospective observational cohort study of 89 English hospital trusts (i.e. groups of hospitals functioning as single operational units). Seven thousand one hundred thirty-three adults admitted to an ICU in England between 2 April and 1 December, 2020 (inclusive), with presumed or confirmed COVID-19, for whom data was submitted to the national surveillance programme and met study inclusion criteria. A Bayesian hierarchical approach was used to model the association between hospital trust level (mechanical ventilation compatible), bed occupancy, and in-hospital all-cause mortality. Results were adjusted for unit characteristics (pre-pandemic size), individual patient-level demographic characteristics (age, sex, ethnicity, deprivation index, time-to-ICU admission), and recorded chronic comorbidities (obesity, diabetes, respiratory disease, liver disease, heart disease, hypertension, immunosuppression, neurological disease, renal disease). RESULTS: One hundred thirty-five thousand six hundred patient days were observed, with a mortality rate of 19.4 per 1000 patient days. Adjusting for patient-level factors, mortality was higher for admissions during periods of high occupancy (> 85% occupancy versus the baseline of 45 to 85%) [OR 1.23 (95% posterior credible interval (PCI): 1.08 to 1.39)]. In contrast, mortality was decreased for admissions during periods of low occupancy (< 45% relative to the baseline) [OR 0.83 (95% PCI 0.75 to 0.94)]. CONCLUSION: Increasing occupancy of beds compatible with mechanical ventilation, a proxy for operational strain, is associated with a higher mortality risk for individuals admitted to ICU
Mateen BA, Wilde H, Dennis JM, et al., 2021, Hospital bed capacity and usage across secondary healthcare providers in England during the first wave of the COVID-19 pandemic: a descriptive analysis, BMJ Open, Vol: 11, Pages: 1-9, ISSN: 2044-6055
Objective In this study, we describe the pattern of bed occupancy across England during the peak of the first wave of the COVID-19 pandemic.Design Descriptive survey.Setting All non-specialist secondary care providers in England from 27 March27to 5 June 2020.Participants Acute (non-specialist) trusts with a type 1 (ie, 24 hours/day, consultant-led) accident and emergency department (n=125), Nightingale (field) hospitals (n=7) and independent sector secondary care providers (n=195).Main outcome measures Two thresholds for ‘safe occupancy’ were used: 85% as per the Royal College of Emergency Medicine and 92% as per NHS Improvement.Results At peak availability, there were 2711 additional beds compatible with mechanical ventilation across England, reflecting a 53% increase in capacity, and occupancy never exceeded 62%. A consequence of the repurposing of beds meant that at the trough there were 8.7% (8508) fewer general and acute beds across England, but occupancy never exceeded 72%. The closest to full occupancy of general and acute bed (surge) capacity that any trust in England reached was 99.8% . For beds compatible with mechanical ventilation there were 326 trust-days (3.7%) spent above 85% of surge capacity and 154 trust-days (1.8%) spent above 92%. 23 trusts spent a cumulative 81 days at 100% saturation of their surge ventilator bed capacity (median number of days per trust=1, range: 1–17). However, only three sustainability and transformation partnerships (aggregates of geographically co-located trusts) reached 100% saturation of their mechanical ventilation beds.Conclusions Throughout the first wave of the pandemic, an adequate supply of all bed types existed at a national level. However, due to an unequal distribution of bed utilisation, many trusts spent a significant period operating above ‘safe-occupancy’ thresholds despite substantial capacity in geographically co-located trusts, a key operational issue to address in pre
Pozharskiy D, Wichrowski NJ, Duncan AB, et al., 2020, Manifold learning for accelerating coarse-grained optimization, Journal of Computational Dynamics, Vol: 7, Pages: 511-536, ISSN: 2158-2505
Algorithms proposed for solving high-dimensional optimization problems with no derivative information frequently encounter the "curse of dimensionality, " becoming ineffective as the dimension of the parameter space grows. One feature of a subclass of such problems that are effectively low-dimensional is that only a few parameters (or combinations thereof) are important for the optimization and must be explored in detail. Knowing these parameters/combinations in advance would greatly simplify the problem and its solution. We propose the data-driven construction of an effective (coarse-grained, "trend") optimizer, based on data obtained from ensembles of brief simulation bursts with an "inner" optimization algorithm, that has the potential to accelerate the exploration of the parameter space. The trajectories of this "effective optimizer" quickly become attracted onto a slow manifold parameterized by the few relevant parameter combinations. We obtain the parameterization of this low-dimensional, effective optimization manifold on the fly using data mining/manifold learning techniques on the results of simulation (inner optimizer iteration) burst ensembles and exploit it locally to "jump" forward along this manifold. As a result, we can bias the exploration of the parameter space towards the few, important directions and, through this "wrapper algorithm, " speed up the convergence of traditional optimization algorithms.
Yates CA, George A, Jordana A, et al., 2020, The blending region hybrid framework for the simulation of stochastic reaction–diffusion processes, Journal of The Royal Society Interface, Vol: 17, Pages: 1-19, ISSN: 1742-5689
The simulation of stochastic reaction–diffusion systems using fine-grained representations can become computationally prohibitive when particle numbers become large. If particle numbers are sufficiently high then it may be possible to ignore stochastic fluctuations and use a more efficient coarse-grained simulation approach. Nevertheless, for multiscale systems which exhibit significant spatial variation in concentration, a coarse-grained approach may not be appropriate throughout the simulation domain. Such scenarios suggest a hybrid paradigm in which a computationally cheap, coarse-grained model is coupled to a more expensive, but more detailed fine-grained model, enabling the accurate simulation of the fine-scale dynamics at a reasonable computational cost. In this paper, in order to couple two representations of reaction–diffusion at distinct spatial scales, we allow them to overlap in a ‘blending region’. Both modelling paradigms provide a valid representation of the particle density in this region. From one end of the blending region to the other, control of the implementation of diffusion is passed from one modelling paradigm to another through the use of complementary ‘blending functions’ which scale up or down the contribution of each model to the overall diffusion. We establish the reliability of our novel hybrid paradigm by demonstrating its simulation on four exemplar reaction–diffusion scenarios.
Seshadri P, Duncan A, Simpson D, et al., 2020, Spatial flow-field approximation using few thermodynamic measurements Part II: Uncertainty assessments, Journal of Turbomachinery
Seshadri P, Simpson D, Thorne G, et al., 2020, Spatial flow-field approximation using few thermodynamic measurements Part I: formulation and area averaging, Journal of Turbomachinery, ISSN: 0889-504X
Our investigation raises an important question that is of relevance to the wider turbomachinery community: howdo we estimate the spatial average of a flow quantity given finite (and sparse) measurements? This paper seeks toadvance efforts to answer this question rigorously. In this paper, we develop a regularized multivariate linear regressionframework for studying engine temperature measurements. As part of this investigation, we study the temperaturemeasurements obtained from the same axial plane across five different engines yielding a total of 82 data-sets. Thefive different engines have similar architectures and therefore similar temperature spatial harmonics are expected. Ourproblem is to estimate the spatial field in engine temperature given a few measurements obtained from thermocouplespositioned on a set of rakes. Our motivation for doing so is to understand key engine temperature modes that cannotbe captured in a rig or in computational simulations, as the cause of these modes may not be replicated in thesesimpler environments. To this end, we develop a multivariate linear least squares model with Tikhonov regularizationto estimate the 2D temperature spatial field. Our model uses a Fourier expansion in the circumferential direction anda quadratic polynomial expansion in the radial direction. One important component of our modeling framework isthe selection of model parameters, i.e. the harmonics in the circumferential direction. A training-testing paradigm isproposed and applied to quantify the harmonics.
Barp A, Briol FX, Duncan A, et al., 2019, Minimum Stein discrepancy estimators, 33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Publisher: Neural Information Processing Systems Foundation, Inc.
When maximum likelihood estimation is infeasible, one often turns to score matching, contrastive divergence, or minimum probability flow to obtain tractable parameter estimates. We provide a unifying perspective of these techniques as minimum Stein discrepancy estimators, and use this lens to design new diffusion kernel Stein discrepancy (DKSD) and diffusion score matching (DSM) estimators with complementary strengths. We establish the consistency, asymptotic normality, and robustness of DKSD and DSM estimators, then derive stochastic Riemannian gradient descent algorithms for their efficient optimisation. The main strength of our methodology is its flexibility, which allows us to design estimators with desirable properties for specific models at hand by carefully selecting a Stein discrepancy. We illustrate this advantage for several challenging problems for score matching, such as non-smooth, heavy-tailed or light-tailed densities.
Gorham J, Duncan A, Vollmer S, et al., 2019, Measuring sample quality with diffusions, Annals of Applied Probability, Vol: 29, Pages: 2884-2928, ISSN: 1050-5164
Stein’s method for measuring convergence to a continuous targetdistribution relies on an operator characterizing the target andSteinfactorbounds on the solutions of an associated differential equation.While such operators and bounds are readily available for a diversityof univariate targets, few multivariate targets have been analyzed. Weintroduce a new class of characterizing operators based on Itˆo diffu-sions and develop explicit multivariate Stein factor bounds for anytarget with a fast-coupling Itˆo diffusion. As example applications, wedevelop computable and convergence-determiningdiffusion Stein dis-crepanciesfor log-concave, heavy-tailed, and multimodal targets anduse these quality measures to select the hyperparameters of biasedMarkov chain Monte Carlo (MCMC) samplers, compare random anddeterministic quadrature rules, and quantify bias-variance tradeoffsin approximate MCMC. Our results establish a near-linear relation-ship between diffusion Stein discrepancies and Wasserstein distances,improving upon past work even for strongly log-concave targets. Theexposed relationship between Stein factors and Markov process cou-pling may be of independent interest.
Duncan A, Zygalakis K, Pavliotis G, 2018, Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation
For a given target density, there exist an infinite number of diffusion processes which are ergodic with respect to this density. As observed in a number of papers, samplers based on nonreversible diffusion processes can significantly outperform their reversible counterparts both in terms of asymptotic variance and rate of convergence to equilibrium. In this paper, we take advantage of this in order to construct efficient sampling algorithms based on the Lie-Trotter decomposition of a nonreversible diffusion process into reversible and nonreversible components. We show that samplers based on this scheme can significantly outperform standard MCMC methods, at the cost of introducing some controlled bias. In particular, we prove that numerical integrators constructed according to this decomposition are geometrically ergodic and characterise fully their asymptotic bias and variance, showing that the sampler inherits the good mixing properties of the underlying nonreversible diffusion. This is illustrated further with a number of numerical examples ranging from highly correlated low dimensional distributions, to logistic regression problems in high dimensions as well as inference for spatial models with many latent variables.
Bierkens J, Bouchard-Côté A, Doucet A, et al., 2018, Piecewise deterministic Markov processes for scalable Monte Carlo on restricted domains, Statistics & Probability Letters, Vol: 136, Pages: 148-154, ISSN: 0167-7152
Duncan AB, Nusken N, Pavliotis GA, 2017, Using perturbed underdamped langevin dynamics to efficiently sample from probability distributions, Journal of Statistical Physics, Vol: 169, Pages: 1098-1131, ISSN: 1572-9613
In this paper we introduce and analyse Langevin samplers that consist of perturbations of the standard underdamped Langevin dynamics. The perturbed dynamics is such that its invariant measure is the same as that of the unperturbed dynamics. We show that appropriate choices of the perturbations can lead to samplers that have improved properties, at least in terms of reducing the asymptotic variance. We present a detailed analysis of the new Langevin sampler for Gaussian target distributions. Our theoretical results are supported by numerical experiments with non-Gaussian target measures.
Bierkens J, Duncan A, 2017, Limit theorems for the zig-zag process, Advances in Applied Probability, Vol: 49, Pages: 791-825, ISSN: 0001-8678
Markov chain Monte Carlo (MCMC) methods provide an essential tool in statistics for sampling from complex probability distributions. While the standard approach to MCMC involves constructing discrete-time reversible Markov chains whose transition kernel is obtained via the Metropolis–Hastings algorithm, there has been recent interest in alternative schemes based on piecewise deterministic Markov processes (PDMPs). One such approach is based on the zig-zag process, introduced in Bierkens and Roberts (2016), which proved to provide a highly scalable sampling scheme for sampling in the big data regime; see Bierkens et al. (2016). In this paper we study the performance of the zig-zag sampler, focusing on the one-dimensional case. In particular, we identify conditions under which a central limit theorem holds and characterise the asymptotic variance. Moreover, we study the influence of the switching rate on the diffusivity of the zig-zag process by identifying a diffusion limit as the switching rate tends to ∞. Based on our results we compare the performance of the zig-zag sampler to existing Monte Carlo methods, both analytically and through simulations.
Kasprzak MJ, Duncan AB, Vollmer SJ, 2017, Note on A. Barbour’s paper on Stein’s method for diffusion approximations, Electronic Communications in Probability, Vol: 22, ISSN: 1083-589X
In [2] foundations for diffusion approximation via Stein’s method are laid. This paper has been cited more than 130 times and is a cornerstone in the area of Stein’s method (see, for example, its use in [1] or [7]). A semigroup argument is used in [2] to solve a Stein equation for Gaussian diffusion approximation. We prove that, contrary to the claim in [2], the semigroup considered therein is not strongly continuous on the Banach space of continuous, real-valued functions on D[0,1] growing slower than a cubic, equipped with an appropriate norm. We also provide a proof of the exact formulation of the solution to the Stein equation of interest, which does not require the aforementioned strong continuity. This shows that the main results of [2] hold true.
Duncan A, Erban R, Zygalakis K, 2016, Hybrid framework for the simulation of stochastic chemical kinetics, Journal of Computational Physics, Vol: 326, Pages: 398-419, ISSN: 0021-9991
Stochasticity plays a fundamental role in various biochemical processes, such as cell regulatory networks and enzyme cascades. Isothermal, well-mixed systems can be modelled as Markov processes, typically simulated using the Gillespie Stochastic Simulation Algorithm (SSA) [25]. While easy to implement and exact, the computational cost of using the Gillespie SSA to simulate such systems can become prohibitive as the frequency of reaction events increases. This has motivated numerous coarse-grained schemes, where the “fast” reactions are approximated either using Langevin dynamics or deterministically. While such approaches provide a good approximation when all reactants are abundant, the approximation breaks down when one or more species exist only in small concentrations and the fluctuations arising from the discrete nature of the reactions become significant. This is particularly problematic when using such methods to compute statistics of extinction times for chemical species, as well as simulating non-equilibrium systems such as cell-cycle models in which a single species can cycle between abundance and scarcity. In this paper, a hybrid jump-diffusion model for simulating well-mixed stochastic kinetics is derived. It acts as a bridge between the Gillespie SSA and the chemical Langevin equation. For low reactant reactions the underlying behaviour is purely discrete, while purely diffusive when the concentrations of all species are large, with the two different behaviours coexisting in the intermediate region. A bound on the weak error in the classical large volume scaling limit is obtained, and three different numerical discretisations of the jump-diffusion model are described. The benefits of such a formalism are illustrated using computational examples.
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