Efficient Large Deviation Estimation Based on Importance Sampling
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Efficient Large Deviation Estimation Based on Importance Sampling Arnaud Guyader1,2 · Hugo Touchette3 Received: 11 March 2020 / Accepted: 12 June 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract We present a complete framework for determining the asymptotic (or logarithmic) efficiency of estimators of large deviation probabilities and rate functions based on importance sampling. The framework relies on the idea that importance sampling in that context is fully characterized by the joint large deviations of two random variables: the observable defining the large deviation probability of interest and the likelihood factor (or Radon–Nikodym derivative) connecting the original process and the modified process used in importance sampling. We recover with this framework known results about the asymptotic efficiency of the exponential tilting and obtain new necessary and sufficient conditions for a general change of process to be asymptotically efficient. This allows us to construct new examples of efficient estimators for sample means of random variables that do not have the exponential tilting form. Other examples involving Markov chains and diffusions are presented to illustrate our results. Keywords Rare events · Importance sampling · Large deviations · Asymptotic efficiency
1 Introduction Estimating the probability of rare events or fluctuations in random systems is an important problem arising in many applied fields, including engineering [1], where a rare event might represent a design failure, or chemistry, where changes between chemical species or polymer states arise from rare transitions in a free energy landscape [2–4]. In physical systems, the probability of rare fluctuations often has a large deviation form [5–8], owing to the interaction
Communicated by Abhishek Dhar.
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Hugo Touchette [email protected]; [email protected] Arnaud Guyader [email protected]
1
Laboratoire de Probabilités, Statistique et Modélisation, Sorbonne Université, Paris, France
2
CERMICS, École des Ponts ParisTech, Champs-sur-Marne, France
3
Department of Mathematical Sciences, Stellenbosch University, Stellenbosch, South Africa
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A. Guyader, H. Touchette
of many particles or the effect of thermal noise. In this case, the estimation of probabilities reduces to the estimation of rate functions, which determine the rate of decay of probabilities as a function of some parameter (e.g., volume, particle number, integration time or temperature) [8]. Rate functions are also important on their own, as they determine for equilibrium and nonequilibrium systems the onset of static and dynamical phase transitions [8–14], fluctuation symmetries [15–18], and in some cases the response to external perturbations [19]. As a result, they have been actively studied recently, especially for nonequilibrium systems describing particle transport processes [20–23] and diffusing particles [24–27], among other physical systems. Traditionally, two statistical methods have been used to numerically estima
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