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 Rd 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) 318–32 9 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 eN as d → ∞ with 1 < eN < N. The convergence is achieved with a computational cost proportional to Nd2. If eN � 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 eN,m as d → ∞ and limm→∞ eN,m = N. Also, we show that the Monte Carlo error for estimating a fixed-dimensional marginal expectation is of order √ 1 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.