On the stability of sequential Monte Carlo methods in high dimensions

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.