An invitation to sequential Monte Carlo samplers

The method of choice for integrating the time-dependent Fokker-Planckequation in high-dimension is to generate samples from the solution viaintegration of the associated stochastic differential equation. Here, we studyan alternative scheme based on integrating an ordinary differential equationthat describes the flow of probability. Acting as a transport map, thisequation deterministically pushes samples from the initial density onto samplesfrom the solution at any later time. Unlike integration of the stochasticdynamics, the method has the advantage of giving direct access to quantitiesthat are challenging to estimate from trajectories alone, such as theprobability current, the density itself, and its entropy. The probability flowequation depends on the gradient of the logarithm of the solution (its"score"), and so is a-priori unknown. To resolve this dependence, we model thescore with a deep neural network that is learned on-the-fly by propagating aset of samples according to the instantaneous probability current. We showtheoretically that the proposed approach controls the Kullback-Leibler divergence from thelearned solution to the target, while learning on external samples from thestochastic differential equation does not control either direction of the Kullback-Leiblerdivergence. Empirically, we consider several high-dimensional Fokker-Planckequations from the physics of interacting particle systems. We find that themethod accurately matches analytical solutions when they are available as wellas moments computed via Monte-Carlo when they are not. Moreover, the methodoffers compelling predictions for the global entropy production rate thatout-perform those obtained from learning on stochastic trajectories, and caneffectively capture non-equilibrium steady-state probability currents over longtime intervals.

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“…Here, we derive some results linking the solution of the transport equation (10) with that of the probability flow equation (6).…”

Section: Data Availability Statementmentioning

confidence: 99%

Probability flow solution of the Fokker–Planck equation

Boffi

Eijnden

2023

Mach. Learn.: Sci. Technol.

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“…We provide here a brief overview of SMC samplers and their connections to AIS. More details can be found in (Del Moral et al, 2006;Dai et al, 2020).…”

Section: Sequential Monte Carlo Samplersmentioning

confidence: 99%

“…As in(Crooks, 1998;Neal, 2001;Del Moral et al, 2006;Dai et al, 2020), we do not use measure-theoretic notation here but it should be kept in mind that the kernels M l do not necessarily admit a density w.r.t. Lebesgue measure; e.g.…”

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confidence: 99%

Annealed Flow Transport Monte Carlo

Arbel,

Matthews,

Doucet

2021

Preprint

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Annealed Importance Sampling (AIS) and its Sequential Monte Carlo (SMC) extensions are stateof-the-art methods for estimating normalizing constants of probability distributions. We propose here a novel Monte Carlo algorithm, Annealed Flow Transport (AFT), that builds upon AIS and SMC and combines them with normalizing flows (NF) for improved performance. This method transports a set of particles using not only importance sampling (IS), Markov chain Monte Carlo (MCMC) and resampling steps -as in SMC, but also relies on NF which are learned sequentially to push particles towards the successive annealed targets. We provide limit theorems for the resulting Monte Carlo estimates of the normalizing constant and expectations with respect to the target distribution. Additionally, we show that a continuous-time scaling limit of the population version of AFT is given by a Feynman-Kac measure which simplifies to the law of a controlled diffusion for expressive NF. We demonstrate experimentally the benefits and limitations of our methodology on a variety of applications.

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“…Compared to MCMC methods or particle MCMC methods (Andrieu, Doucet, and Holenstein, 2010) that sample from the posterior distribution using a single Markov chain, our algorithm shares the many advantages of SMC samplers (Del Moral et al, 2006;Dai et al, 2020). This includes parallelism over the parameter particles, automated tuning of the inverse temperatures and proposal transitions in the parameter space, and an estimator of the model evidence that facilitiates model comparison.…”

Section: Introductionmentioning

confidence: 99%

Efficient Likelihood-based Estimation via Annealing for Dynamic Structural Macrofinance Models

Fülöp¹,

Heng²,

Li³

2022

Preprint

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Most solved dynamic structural macrofinance models are non-linear and/or non-Gaussian state-space models with high-dimensional and complex structures. We propose an annealed controlled sequential Monte Carlo method that delivers numerically stable and low variance estimators of the likelihood function. The method relies on an annealing procedure to gradually introduce information from observations and constructs globally optimal proposal distributions by solving associated optimal control problems that yield zero variance likelihood estimators. To perform parameter inference, we develop a new adaptive SMC 2 algorithm that employs likelihood estimators from annealed controlled sequential Monte Carlo. We provide a theoretical stability analysis that elucidates the advantages of our methodology and asymptotic results concerning the consistency and convergence rates of our SMC 2 estimators. We illustrate the strengths of our proposed methodology by estimating two popular macrofinance models: a non-linear new Keynesian dynamic stochastic general equilibrium model and a nonlinear non-Gaussian consumption-based long-run risk model.

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An invitation to sequential Monte Carlo samplers

Cited by 4 publications

References 108 publications

Probability flow solution of the Fokker–Planck equation

Probability flow solution of the Fokker–Planck equation

Annealed Flow Transport Monte Carlo

Efficient Likelihood-based Estimation via Annealing for Dynamic Structural Macrofinance Models

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