A piecewise deterministic Monte Carlo method for diffusion bridges

Bierkens, Joris; Grazzi, Sebastiano; Meulen, Frank van der; Schauer, Moritz

doi:10.1007/s11222-021-10008-8

Cited by 17 publications

(24 citation statements)

References 23 publications

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“…Let t m be the maximum time it takes for a flow that starts from inside O m and solves (8) to exit O m . For any m, on the event {E n ≥ λt m }, if the process has not escaped the ball O m until the n − 1'th switch, it does so following the dynamics before the n'th switch happens.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 99%

“…In [6], the authors also introduce some variants of the algorithm that use the technique of subsampling, improving computational efficiency when the target distribution is obtained from a Bayesian analysis involving a large data set. Further literature on the topic includes [5,4,9,12,8,7]. [11] proves ergodicity and exponential ergodicity of the Zig-Zag process in arbitrary dimension.…”

Section: Introductionmentioning

confidence: 99%

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Speed Up Zig-Zag

Vasdekis¹,

Roberts²

2021

Preprint

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Zig-Zag is Piecewise Deterministic Markov Process, efficiently used for simulation in an MCMC setting. As we show in this article, it fails to be exponentially ergodic on heavy tailed target distributions. We introduce an extension of the Zig-Zag process by allowing the process to move with a non-constant speed function s, depending on the current state of the process. We call this process Speed Up Zig-Zag (SUZZ). We provide conditions that guarantee stability properties for the SUZZ process, including non-explosivity, exponential ergodicity in heavy tailed targets and central limit theorem. Interestingly, we find that using speed functions that induce explosive deterministic dynamics may lead to stable algorithms that can even mix faster. We further discuss the choice of an efficient speed function by providing an efficiency criterion for the one-dimensional process and we support our findings with simulation results.

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Section: Proofs Of the Main Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Speed Up Zig-Zag

Vasdekis¹,

Roberts²

2021

Preprint

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“…The sparse conditional dependence structure implies that the individual switching intensities λ i (x) are in fact functions of a subset of the components of x, contributing to a fast computation. This feature can be exploited by an efficient 'local' implementation of the FBS algorithm which reduces the number of Poisson times simulated by the algorithm (similar in spirit to the local Bouncy Particle Sampler (Bouchard-Côté et al, 2017) and the local Zig-Zag sampler in (Bierkens et al, 2020)). In Section 3.2 we will briefly comment on the dimensional scaling of FBS.…”

Section: Factorised Boomerang Samplermentioning

confidence: 99%

“…In (Bierkens et al, 2020) the authors introduce a framework for the simulation of diffusion bridges (diffusion processes conditioned to hit a prescribed endpoint) taking strong advantage of the use of factorised piecewise deterministic samplers. This invites the use of the Factorised Boomerang Sampler (FBS).…”

Section: Diffusion Bridgesmentioning

confidence: 99%

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The Boomerang Sampler

Bierkens,

Grazzi,

Kamatani

et al. 2020

Preprint

Self Cite

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This paper introduces the boomerang sampler as a novel class of continuous-time non-reversible Markov chain Monte Carlo algorithms. The methodology begins by representing the target density as a density, e −U , with respect to a prescribed (usually) Gaussian measure and constructs a continuous trajectory consisting of a piecewise elliptical path. The method moves from one elliptical orbit to another according to a rate function which can be written in terms of U . We demonstrate that the method is easy to implement and demonstrate empirically that it can out-perform existing benchmark piecewise deterministic Markov processes such as the bouncy particle sampler and the Zig-Zag. In the Bayesian statistics context, these competitor algorithms are of substantial interest in the large data context due to the fact that they can adopt data subsampling techniques which are exact (ie induce no error in the stationary distribution). We demonstrate theoretically and empirically that we can also construct a control-variate subsampling boomerang sampler which is also exact, and which possesses remarkable scaling properties in the large data limit. We furthermore illustrate a factorised version on the simulation of diffusion bridges.1. We relax a condition which restricts the covariance function of the auxiliary velocity process to be isotropic. This generalisation is crucial to ensure good convergence properties of the algorithm.2. Furthermore we extend the Boomerang Sampler to allow for exact subsampling (as introduced above),

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Statistical inference for stochastic differential equations

Craigmile

Herbei

Liu

et al. 2022

WIREs Computational Stats

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Many scientific fields have experienced growth in the use of stochastic differential equations (SDEs), also known as diffusion processes, to model scientific phenomena over time. SDEs can simultaneously capture the known deterministic dynamics of underlying variables of interest (e.g., ocean flow, chemical and physical characteristics of a body of water, presence, absence, and spread of a disease), while enabling a modeler to capture the unknown random dynamics in a stochastic setting. We focus on reviewing a wide range of statistical inference methods for likelihood-based frequentist and Bayesian parametric inference based on discretely-sampled diffusions. Exact parametric inference is not usually possible because the transition density is not available in closed form. Thus, we review the literature on approximate numerical methods (e.g., Euler, Milstein, local linearization, and Aït-Sahalia) and simulation-based approaches (e.g., data augmentation and exact sampling) that are used to carry out parametric statistical inference on SDE processes. We close with a brief discussion of other methods of inference for SDEs and more complex SDE processes such as spatio-temporal SDEs.

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A piecewise deterministic Monte Carlo method for diffusion bridges

Cited by 17 publications

References 23 publications

Speed Up Zig-Zag

Speed Up Zig-Zag

The Boomerang Sampler

Statistical inference for stochastic differential equations

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