Multilevel sequential Monte Carlo samplers

Beskos, Alexandros; Jasra, Ajay; Law, Kody J. H.; Tempone, Raúl; Zhou, Yan

doi:10.1016/j.spa.2016.08.004

Cited by 111 publications

(172 citation statements)

References 18 publications

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“…More recently, it has been shown that the MLMC telescoping summation can accelerate the computation of posterior expectations when using MCMC [29,32] or SMC [12].…”

Section: Calculate Acceptance Probabilitymentioning

confidence: 99%

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Warne

Baker

Simpson

2019

J. R. Soc. Interface.

112

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Stochasticity is a key characteristic of intracellular processes such as gene regulation and chemical signalling. Therefore, characterizing stochastic effects in biochemical systems is essential to understand the complex dynamics of living things. Mathematical idealizations of biochemically reacting systems must be able to capture stochastic phenomena. While robust theory exists to describe such stochastic models, the computational challenges in exploring these models can be a significant burden in practice since realistic models are analytically intractable. Determining the expected behaviour and variability of a stochastic biochemical reaction network requires many probabilistic simulations of its evolution. Using a biochemical reaction network model to assist in the interpretation of time-course data from a biological experiment is an even greater challenge due to the intractability of the likelihood function for determining observation probabilities. These computational challenges have been subjects of active research for over four decades. In this review, we present an accessible discussion of the major historical developments and state-of-the-art computational techniques relevant to simulation and inference problems for stochastic biochemical reaction network models. Detailed algorithms for particularly important methods are described and complemented with Matlab ® implementations. As a result, this review provides a practical and accessible introduction to computational methods for stochastic models within the life sciences community.

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“…More recently, it has been shown that the MLMC telescoping summation can accelerate the computation of posterior expectations when using MCMC [29,32] or SMC [12].…”

Section: Calculate Acceptance Probabilitymentioning

confidence: 99%

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Warne

Baker

Simpson

2019

J. R. Soc. Interface.

112

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“…For the solution of Bayesian inverse problems, there are multi-stage MCMC methods that aim to reduce the number of high-fidelity model evaluations by first screening proposed moves with lowfidelity models [14,22]. Another line of work builds on hierarchies of low-fidelity models, typically derived from different discretizations of partial differential equations (PDEs) underlying the high-fidelity model, to reduce sampling costs [4,20,31].…”

Section: Introductionmentioning

confidence: 99%

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Peherstorfer

Marzouk

2019

Adv Comput Math

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Markov chain Monte Carlo (MCMC) sampling of posterior distributions arising in Bayesian inverse problems is challenging when evaluations of the forward model are computationally expensive. Replacing the forward model with a low-cost, low-fidelity model often significantly reduces computational cost; however, employing a low-fidelity model alone means that the stationary distribution of the MCMC chain is the posterior distribution corresponding to the low-fidelity model, rather than the original posterior distribution corresponding to the high-fidelity model. We propose a multifidelity approach that combines, rather than replaces, the high-fidelity model with a low-fidelity model. First, the low-fidelity model is used to construct a transport map that deterministically couples a reference Gaussian distribution with an approximation of the low-fidelity posterior. Then, the high-fidelity posterior distribution is explored using a non-Gaussian proposal distribution derived from the transport map. This multifidelity "preconditioned" MCMC approach seeks efficient sampling via a proposal that is explicitly tailored to the posterior at hand and that is constructed efficiently with the low-fidelity model. By relying on the low-fidelity model only to construct the proposal distribution, our approach guarantees that the stationary distribution of the MCMC chain is the high-fidelity posterior. In our numerical examples, our multifidelity approach achieves significant speedups compared to single-fidelity MCMC sampling methods.Keywords: Bayesian inverse problems; transport maps; multifidelity; model reduction; Markov chain Monte Carlo * The second author acknowledges support of the AFOSR MURI on multi-information sources of multi-physics systems under Award Number FA9550-15-1-0038.

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“…For Bayesian Inverse Problems a multilevel MCMC method has been introduced in [18]. Multilevel Sequential Monte Carlo is introduced in [4] and further discussed in [3,17,16]. Both the multilevel MCMC and multilevel SMC use coarse PDE discretisations for variance reduction with the help of a telescoping sum expansion.…”

Section: Introductionmentioning

confidence: 99%

“…This update scheme is consistent with the adaptive bridging and tempering discussed in §3. 4. Finally, in §5 we present numerical experiments for a test problem in 2D space.…”

Section: Introductionmentioning

confidence: 99%

Multilevel Sequential2 Monte Carlo for Bayesian inverse problems

Latz

Papaioannou²,

Ullmann

2018

Journal of Computational Physics

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The identification of parameters in mathematical models using noisy observations is a common task in uncertainty quantification. We employ the framework of Bayesian inversion: we combine monitoring and observational data with prior information to estimate the posterior distribution of a parameter. Specifically, we are interested in the distribution of a diffusion coefficient of an elliptic PDE. In this setting, the sample space is high-dimensional, and each sample of the PDE solution is expensive. To address these issues we propose and analyse a novel Sequential Monte Carlo (SMC) sampler for the approximation of the posterior distribution. Classical, single-level SMC constructs a sequence of measures, starting with the prior distribution, and finishing with the posterior distribution. The intermediate measures arise from a tempering of the likelihood, or, equivalently, a rescaling of the noise. The resolution of the PDE discretisation is fixed. In contrast, our estimator employs a hierarchy of PDE discretisations to decrease the computational cost. We construct a sequence of intermediate measures by decreasing the temperature or by increasing the discretisation level at the same time. This idea builds on and generalises the multi-resolution sampler proposed in [P.S. Koutsourelakis, J. Comput. Phys., 228 (2009), pp. 6184-6211] where a bridging scheme is used to transfer samples from coarse to fine discretisation levels. Importantly, our choice between tempering and bridging is fully adaptive. We present numerical experiments in 2D space, comparing our estimator to single-level SMC and the multi-resolution sampler.In contrast to deterministic regularisation techniques, the Bayesian approach to inverse problems uses the probabilistic framework of Bayesian inference. Bayesian inference is built on Bayes' Formula in the formulation given by Laplace [34, II.1]. We remark that other formulations are possible, see e.g. the work by Matthies et al. [38]. We make use of the mathematical framework for Bayesian Inverse Problems (BIPs) given by Stuart [48]. Under weak assumptions -which we will give below -one can show that the BIP is well-posed. The solution of the BIP is the conditional probability measure of the unknown parameter given the observations.The Bayesian framework is very general and can handle different types of forward models. However, in this work we consider PDE-based forward models, and in particular an elliptic PDE. The exact solution of the associated BIP is often inaccessible for two reasons: (i) there is no closed form expression for the posterior measure, and (ii) the underlying PDE cannot be solved analytically. We focus on (i), and study efficient approximations to the full posterior measure. Alternatively, one could also only approximate the expectation of output quantities of interest with respect to the posterior measure, or estimate the model evidence, the normalization constant of the posterior measure.Typically, BIPs are approached with sampling based methods, such as Markov Chain Monte Carlo (M...

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Multilevel sequential Monte Carlo samplers

Cited by 111 publications

References 18 publications

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

Simulation and inference algorithms for stochastic biochemical reaction networks: from basic concepts to state-of-the-art

A transport-based multifidelity preconditioner for Markov chain Monte Carlo

Multilevel Sequential2 Monte Carlo for Bayesian inverse problems

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