arXiv subjects

Andrew Golightly

Publications and source records attributed to Andrew Golightly.

32 records · Page 2Linked to original sources

arXiv2019

Efficient sampling of conditioned Markov jump processes

We consider the task of generating draws from a Markov jump process (MJP) between two time-points at which the process is known. Resulting draws are typically termed bridges and the generation of such bridges plays a key role in simulation-based inference algorithms for MJPs. The problem is challenging due to the intractability of the conditioned process, necessitating the use of computationally intensive methods such as weighted resampling or Markov chain Monte Carlo. An efficient implementation of such schemes requires an approximation of the intractable conditioned hazard/propensity function that is both cheap and accurate. In this paper, we review some existing approaches to this problem before outlining our novel contribution. Essentially, we leverage the tractability of a Gaussian approximation of the MJP and suggest a computationally efficient implementation of the resulting conditioned hazard approximation. We compare and contrast our approach with existing methods using three examples.

stat.CO↗

arXiv2019

Correlated pseudo-marginal schemes for time-discretised stochastic kinetic models

The challenging problem of conducting fully Bayesian inference for the reaction rate constants governing stochastic kinetic models (SKMs) is considered. Given the challenges underlying this problem, the Markov jump process representation is routinely replaced by an approximation based on a suitable time discretisation of the system of interest. Improving the accuracy of these schemes amounts to using an ever finer discretisation level, which in the context of the inference problem, requires integrating over the uncertainty in the process at a predetermined number of intermediate times between observations. Pseudo-marginal Metropolis-Hastings schemes are increasingly used, since for a given discretisation level, the observed data likelihood can be unbiasedly estimated using a particle filter. When observations are particularly informative an auxiliary particle filter can be implemented, by employing an appropriate construct to push the state particles towards the observations in a sensible way. Recent work in state-space settings has shown how the pseudo-marginal approach can be made much more efficient by correlating the underlying pseudo-random numbers used to form the likelihood estimate at the current and proposed values of the unknown parameters. We extend this approach to the time-discretised SKM framework by correlating the innovations that drive the auxiliary particle filter. We find that the resulting approach offers substantial gains in efficiency over a standard implementation.

stat.CO↗

arXiv2018

Sequential Bayesian inference for spatio-temporal models of temperature and humidity data

We develop a spatio-temporal model to forecast sensor output at five locations in North East England. The signal is described using coupled dynamic linear models, with spatial effects specified by a Gaussian process. Data streams are analysed using a stochastic algorithm which sequentially approximates the parameter posterior through a series of reweighting and resampling steps. An iterated batch importance sampling scheme is used to circumvent particle degeneracy through a resample-move step. The algorithm is modified to make it more efficient and parallisable. The model is shown to give a good description of the underlying process and provide reasonable forecast accuracy.

stat.AP↗

arXiv2018

Black-box Variational Inference for Stochastic Differential Equations

Parameter inference for stochastic differential equations is challenging due to the presence of a latent diffusion process. Working with an Euler-Maruyama discretisation for the diffusion, we use variational inference to jointly learn the parameters and the diffusion paths. We use a standard mean-field variational approximation of the parameter posterior, and introduce a recurrent neural network to approximate the posterior for the diffusion paths conditional on the parameters. This neural network learns how to provide Gaussian state transitions which bridge between observations in a very similar way to the conditioned diffusion process. The resulting black-box inference method can be applied to any SDE system with light tuning requirements. We illustrate the method on a Lotka-Volterra system and an epidemic model, producing accurate parameter estimates in a few hours.

stat.CO↗

arXiv2017

Efficient SMC$^2$ schemes for stochastic kinetic models

Fitting stochastic kinetic models represented by Markov jump processes within the Bayesian paradigm is complicated by the intractability of the observed data likelihood. There has therefore been considerable attention given to the design of pseudo-marginal Markov chain Monte Carlo algorithms for such models. However, these methods are typically computationally intensive, often require careful tuning and must be restarted from scratch upon receipt of new observations. Sequential Monte Carlo (SMC) methods on the other hand aim to efficiently reuse posterior samples at each time point. Despite their appeal, applying SMC schemes in scenarios with both dynamic states and static parameters is made difficult by the problem of particle degeneracy. A principled approach for overcoming this problem is to move each parameter particle through a Metropolis-Hastings kernel that leaves the target invariant. This rejuvenation step is key to a recently proposed SMC$^2$ algorithm, which can be seen as the pseudo-marginal analogue of an idealised scheme known as iterated batch importance sampling. Computing the parameter weights in SMC$^2$ requires running a particle filter over dynamic states to unbiasedly estimate the intractable observed data likelihood contributions at each time point. In this paper, we propose to use an auxiliary particle filter inside the SMC$^2$ scheme. Our method uses two recently proposed constructs for sampling conditioned jump processes and we find that the resulting inference schemes typically require fewer state particles than when using a simple bootstrap filter. Using two applications, we compare the performance of the proposed approach with various competing methods, including two global MCMC schemes.

stat.CO↗

arXiv2016

Diagnostics for assessing the linear noise and moment closure approximations

Solving the chemical master equation exactly is typically not possible, so instead we must rely on simulation based methods. Unfortunately, drawing exact realisations, results in simulating every reaction that occurs. This will preclude the use of exact simulators for models of any realistic size and so approximate algorithms become important. In this paper we describe a general framework for assessing the accuracy of the linear noise and two moment approximations. By constructing an efficient space filling design over the parameter region of interest, we present a number of useful diagnostic tools that aids modellers in assessing whether the approximation is suitable. In particular, we leverage the normality assumption of the linear noise and moment closure approximations.

q-bio.QM↗

arXiv2016

Adaptive, delayed-acceptance MCMC for targets with expensive likelihoods

When conducting Bayesian inference, delayed acceptance (DA) Metropolis-Hastings (MH) algorithms and DA pseudo-marginal MH algorithms can be applied when it is computationally expensive to calculate the true posterior or an unbiased estimate thereof, but a computationally cheap approximation is available. A first accept-reject stage is applied, with the cheap approximation substituted for the true posterior in the MH acceptance ratio. Only for those proposals which pass through the first stage is the computationally expensive true posterior (or unbiased estimate thereof) evaluated, with a second accept-reject stage ensuring that detailed balance is satisfied with respect to the intended true posterior. In some scenarios there is no obvious computationally cheap approximation. A weighted average of previous evaluations of the computationally expensive posterior provides a generic approximation to the posterior. If only the $k$-nearest neighbours have non-zero weights then evaluation of the approximate posterior can be made computationally cheap provided that the points at which the posterior has been evaluated are stored in a multi-dimensional binary tree, known as a KD-tree. The contents of the KD-tree are potentially updated after every computationally intensive evaluation. The resulting adaptive, delayed-acceptance [pseudo-marginal] Metropolis-Hastings algorithm is justified both theoretically and empirically. Guidance on tuning parameters is provided and the methodology is applied to a discretely observed Markov jump process characterising predator-prey interactions and an ODE system describing the dynamics of an autoregulatory gene network.

stat.CO↗

arXiv2016

Bayesian inference for diffusion driven mixed-effects models

Stochastic differential equations (SDEs) provide a natural framework for modelling intrinsic stochasticity inherent in many continuous-time physical processes. When such processes are observed in multiple individuals or experimental units, SDE driven mixed-effects models allow the quantification of between (as well as within) individual variation. Performing Bayesian inference for such models, using discrete time data that may be incomplete and subject to measurement error is a challenging problem and is the focus of this paper. We extend a recently proposed MCMC scheme to include the SDE driven mixed-effects framework. Fundamental to our approach is the development of a novel construct that allows for efficient sampling of conditioned SDEs that may exhibit nonlinear dynamics between observation times. We apply the resulting scheme to synthetic data generated from a simple SDE model of orange tree growth, and real data consisting of observations on aphid numbers recorded under a variety of different treatment regimes. In addition, we provide a systematic comparison of our approach with an inference scheme based on a tractable approximation of the SDE, that is, the linear noise approximation.

stat.CO↗

arXiv2016

Improved bridge constructs for stochastic differential equations

We consider the task of generating discrete-time realisations of a nonlinear multivariate diffusion process satisfying an Itô stochastic differential equation conditional on an observation taken at a fixed future time-point. Such realisations are typically termed diffusion bridges. Since, in general, no closed form expression exists for the transition densities of the process of interest, a widely adopted solution works with the Euler-Maruyama approximation, by replacing the intractable transition densities with Gaussian approximations. However, the density of the conditioned discrete-time process remains intractable, necessitating the use of computationally intensive methods such as Markov chain Monte Carlo. Designing an efficient proposal mechanism which can be applied to a noisy and partially observed system that exhibits nonlinear dynamics is a challenging problem, and is the focus of this paper. By partitioning the process into two parts, one that accounts for nonlinear dynamics in a deterministic way, and another as a residual stochastic process, we develop a class of novel constructs that bridge the residual process via a linear approximation. In addition, we adapt a recently proposed construct to a partial and noisy observation regime. We compare the performance of each new construct with a number of existing approaches, using three applications.

stat.CO↗

arXiv2014

Bayesian inference for Markov jump processes with informative observations

In this paper we consider the problem of parameter inference for Markov jump process (MJP) representations of stochastic kinetic models. Since transition probabilities are intractable for most processes of interest yet forward simulation is straightforward, Bayesian inference typically proceeds through computationally intensive methods such as (particle) MCMC. Such methods ostensibly require the ability to simulate trajectories from the conditioned jump process. When observations are highly informative, use of the forward simulator is likely to be inefficient and may even preclude an exact (simulation based) analysis. We therefore propose three methods for improving the efficiency of simulating conditioned jump processes. A conditioned hazard is derived based on an approximation to the jump process, and used to generate end-point conditioned trajectories for use inside an importance sampling algorithm. We also adapt a recently proposed sequential Monte Carlo scheme to our problem. Essentially, trajectories are reweighted at a set of intermediate time points, with more weight assigned to trajectories that are consistent with the next observation. We consider two implementations of this approach, based on two continuous approximations of the MJP. We compare these constructs for a simple tractable jump process before using them to perform inference for a Lotka-Volterra system. The best performing construct is used to infer the parameters governing a simple model of motility regulation in Bacillus subtilis.

stat.CO↗

arXiv2014

Delayed acceptance particle MCMC for exact inference in stochastic kinetic models

Recently-proposed particle MCMC methods provide a flexible way of performing Bayesian inference for parameters governing stochastic kinetic models defined as Markov (jump) processes (MJPs). Each iteration of the scheme requires an estimate of the marginal likelihood calculated from the output of a sequential Monte Carlo scheme (also known as a particle filter). Consequently, the method can be extremely computationally intensive. We therefore aim to avoid most instances of the expensive likelihood calculation through use of a fast approximation. We consider two approximations: the chemical Langevin equation diffusion approximation (CLE) and the linear noise approximation (LNA). Either an estimate of the marginal likelihood under the CLE, or the tractable marginal likelihood under the LNA can be used to calculate a first step acceptance probability. Only if a proposal is accepted under the approximation do we then run a sequential Monte Carlo scheme to compute an estimate of the marginal likelihood under the true MJP and construct a second stage acceptance probability that permits exact (simulation based) inference for the MJP. We therefore avoid expensive calculations for proposals that are likely to be rejected. We illustrate the method by considering inference for parameters governing a Lotka-Volterra system, a model of gene expression and a simple epidemic process.

stat.CO↗

arXiv2014

Bayesian Inference for Hybrid Discrete-Continuous Stochastic Kinetic Models

We consider the problem of efficiently performing simulation and inference for stochastic kinetic models. Whilst it is possible to work directly with the resulting Markov jump process, computational cost can be prohibitive for networks of realistic size and complexity. In this paper, we consider an inference scheme based on a novel hybrid simulator that classifies reactions as either "fast" or "slow" with fast reactions evolving as a continuous Markov process whilst the remaining slow reaction occurrences are modelled through a Markov jump process with time dependent hazards. A linear noise approximation (LNA) of fast reaction dynamics is employed and slow reaction events are captured by exploiting the ability to solve the stochastic differential equation driving the LNA. This simulation procedure is used as a proposal mechanism inside a particle MCMC scheme, thus allowing Bayesian inference for the model parameters. We apply the scheme to a simple application and compare the output with an existing hybrid approach and also a scheme for performing inference for the underlying discrete stochastic model.

stat.CO↗

arXiv2013

Inference for population dynamics in the Neolithic period

We consider parameter estimation for the spread of the Neolithic incipient farming across Europe using radiocarbon dates. We model the arrival time of farming at radiocarbon-dated, early Neolithic sites by a numerical solution to an advancing wavefront. We allow for (technical) uncertainty in the radiocarbon data, lack-of-fit of the deterministic model and use a Gaussian process to smooth spatial deviations from the model. Inference for the parameters in the wavefront model is complicated by the computational cost required to produce a single numerical solution. We therefore employ Gaussian process emulators for the arrival time of the advancing wavefront at each radiocarbon-dated site. We validate our model using predictive simulations.

stat.AP↗

arXiv2012

Bayesian inference for a wavefront model of the Neolithisation of Europe

We consider a wavefront model for the spread of Neolithic culture across Europe, and use Bayesian inference techniques to provide estimates for the parameters within this model, as constrained by radiocarbon data from Southern and Western Europe. Our wavefront model allows for both an isotropic background spread (incorporating the effects of local geography), and a localized anisotropic spread associated with major waterways. We introduce an innovative numerical scheme to track the wavefront, and use Gaussian process emulators to further increase the efficiency of our model, thereby making Markov chain Monte Carlo methods practical. We allow for uncertainty in the fit of our model, and discuss the inferred distribution of the parameter specifying this uncertainty, along with the distributions of the parameters of our wavefront model. We subsequently use predictive distributions, taking account of parameter uncertainty, to identify radiocarbon sites which do not agree well with our model. These sites may warrant further archaeological study, or motivate refinements to the model.

physics.bio-ph↗