Imperial College London
Alternate

Dr Nikolas Kantas

Faculty of Natural Sciences, Department of Mathematics

Reader in Statistics
 
 
 
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Contact

 

+44 (0)20 7594 2772n.kantas Website

 
 
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Location

 

538Huxley BuildingSouth Kensington Campus

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Summary

 

Publications

Citation

BibTex format

@article{Ruzayqat:2023:10.1016/j.jcp.2022.111643,
author = {Ruzayqat, H and Beskos, A and Crisan, D and Jasra, A and Kantas, N},
doi = {10.1016/j.jcp.2022.111643},
journal = {Journal of Computational Physics},
title = {Unbiased estimation using a class of diffusion processes},
url = {http://dx.doi.org/10.1016/j.jcp.2022.111643},
volume = {472},
year = {2023}
}
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RIS format (EndNote, RefMan)

TY  - JOUR
AB  - We study the problem of unbiased estimation of expectations with respect to(w.r.t.) $\pi$ a given, general probability measure on$(\mathbb{R}^d,\mathcal{B}(\mathbb{R}^d))$ that is absolutely continuous withrespect to a standard Gaussian measure. We focus on simulation associated to aparticular class of diffusion processes, sometimes termed theSchr\"odinger-F\"ollmer Sampler, which is a simulation technique thatapproximates the law of a particular diffusion bridge process $\{X_t\}_{t\in[0,1]}$ on $\mathbb{R}^d$, $d\in \mathbb{N}_0$. This latter process isconstructed such that, starting at $X_0=0$, one has $X_1\sim \pi$. Typically,the drift of the diffusion is intractable and, even if it were not, exactsampling of the associated diffusion is not possible. As a result,\cite{sf_orig,jiao} consider a stochastic Euler-Maruyama scheme that allows thedevelopment of biased estimators for expectations w.r.t.~$\pi$. We show thatfor this methodology to achieve a mean square error of$\mathcal{O}(\epsilon^2)$, for arbitrary $\epsilon>0$, the associated cost is$\mathcal{O}(\epsilon^{-5})$. We then introduce an alternative approach thatprovides unbiased estimates of expectations w.r.t.~$\pi$, that is, it does notsuffer from the time discretization bias or the bias related with theapproximation of the drift function. We prove that to achieve a mean squareerror of $\mathcal{O}(\epsilon^2)$, the associated cost is, with highprobability, $\mathcal{O}(\epsilon^{-2}|\log(\epsilon)|^{2+\delta})$, for any$\delta>0$. We implement our method on several examples including Bayesianinverse problems.
AU  - Ruzayqat,H
AU  - Beskos,A
AU  - Crisan,D
AU  - Jasra,A
AU  - Kantas,N
DO  - 10.1016/j.jcp.2022.111643
PY  - 2023///
SN  - 0021-9991
TI  - Unbiased estimation using a class of diffusion processes
T2  - Journal of Computational Physics
UR  - http://dx.doi.org/10.1016/j.jcp.2022.111643
UR  - http://arxiv.org/abs/2203.03013v2
UR  - http://hdl.handle.net/10044/1/100101
VL  - 472
ER  -
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