The Annals of Applied Probability

On the stability of sequential Monte Carlo methods in high dimensions

Alexandros Beskos, Dan Crisan, and Ajay Jasra

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Abstract

We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on $\mathbb{R}^{d}$ for large $d$. It is well known [Bengtsson, Bickel and Li, In Probability and Statistics: Essays in Honor of David A. Freedman, D. Nolan and T. Speed, eds. (2008) 316–334 IMS; see also Pushing the Limits of Contemporary Statistics (2008) 318–329 IMS, Mon. Weather Rev. (2009) 136 (2009) 4629–4640] that using a single importance sampling step, one produces an approximation for the target that deteriorates as the dimension $d$ increases, unless the number of Monte Carlo samples $N$ increases at an exponential rate in $d$. We show that this degeneracy can be avoided by introducing a sequence of artificial targets, starting from a “simple” density and moving to the one of interest, using an SMC method to sample from the sequence; see, for example, Chopin [Biometrika 89 (2002) 539–551]; see also [J. R. Stat. Soc. Ser. B Stat. Methodol. 68 (2006) 411–436, Phys. Rev. Lett. 78 (1997) 2690–2693, Stat. Comput. 11 (2001) 125–139]. Using this class of SMC methods with a fixed number of samples, one can produce an approximation for which the effective sample size (ESS) converges to a random variable $\varepsilon_{N}$ as $d\rightarrow\infty$ with $1<\varepsilon_{N}<N$. The convergence is achieved with a computational cost proportional to $Nd^{2}$. If $\varepsilon_{N}\ll N$, we can raise its value by introducing a number of resampling steps, say $m$ (where $m$ is independent of $d$). In this case, the ESS converges to a random variable $\varepsilon_{N,m}$ as $d\rightarrow\infty$ and $\lim_{m\to\infty}\varepsilon_{N,m}=N$. Also, we show that the Monte Carlo error for estimating a fixed-dimensional marginal expectation is of order $\frac{1}{\sqrt{N}}$ uniformly in $d$. The results imply that, in high dimensions, SMC algorithms can efficiently control the variability of the importance sampling weights and estimate fixed-dimensional marginals at a cost which is less than exponential in $d$ and indicate that resampling leads to a reduction in the Monte Carlo error and increase in the ESS. All of our analysis is made under the assumption that the target density is i.i.d.

Article information

Source
Ann. Appl. Probab. Volume 24, Number 4 (2014), 1396-1445.

Dates
First available in Project Euclid: 14 May 2014

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1400073653

Digital Object Identifier
doi:10.1214/13-AAP951

Mathematical Reviews number (MathSciNet)
MR3211000

Zentralblatt MATH identifier
1304.82070

Subjects
Primary: 82C80: Numerical methods (Monte Carlo, series resummation, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60F99: None of the above, but in this section 62F15: Bayesian inference

Keywords
Sequential Monte Carlo high dimensions resampling functional CLT

Citation

Beskos, Alexandros; Crisan, Dan; Jasra, Ajay. On the stability of sequential Monte Carlo methods in high dimensions. Ann. Appl. Probab. 24 (2014), no. 4, 1396--1445. doi:10.1214/13-AAP951. https://projecteuclid.org/euclid.aoap/1400073653.


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