ON THE STABILITY OF SEQUENTIAL MONTE CARLO METHODS IN HIGH DIMENSIONS
ON THE STABILITY OF SEQUENTIAL MONTE CARLO METHODS IN HIGH DIMENSIONS
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DOI：
10.1214/13-aap951
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发表时间：
2014-08-01
影响因子：
1.8
通讯作者：
Jasra, Ajay
中科院分区：
文献类型：
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作者：
Beskos, Alexandros;Crisan, Dan;Jasra, Ajay
我们研究了序贯蒙特卡罗 (SMC) 方法的稳定性，该方法应用于从大 d 的 R-d 上的目标分布进行采样的问题。这是众所周知的[Bengtsson、Bickel 和 Li，《概率与统计：纪念 David A. Freedman、D. Nolan 和 T. Speed》编辑的论文。 (2008) 316-334 IMS；另请参阅《推动当代统计的极限》(2008) 318-329 IMS，周一。 Weather Rev. (2009) 136 (2009) 4629-4640] 使用单个重要性采样步骤，可以生成随着维度 d 增加而恶化的目标近似值，除非蒙特卡罗样本 N 的数量以 d 中的指数速率增加。我们证明，可以通过引入一系列人工目标来避免这种简并性，从“简单”密度开始，移动到感兴趣的密度，使用 SMC 方法从序列中采样；参见例如肖邦[Biometrika 89 (2002) 539-551]；另见[J. R. 统计。苏克。序列。 B 统计数据。方法。 68 (2006) 411-436，物理。莱特牧师。 78 (1997) 2690-2693，统计数据。计算。 11（2001）125-139]。使用此类具有固定数量样本的 SMC 方法，可以生成一个近似值，使有效样本大小 (ESS) 收敛到随机变量 epsilon(N)，即 d -> 无穷大且 1 < epsilon(N) < N。通过与 Nd-2 成比例的计算成本实现收敛。如果 epsilon(N) 无穷大且 lim(m ->infinity) epsilon(N,m) = N。此外，我们还表明，用于估计固定维边际期望的蒙特卡洛误差在 d 中统一为 1/根 N。结果表明，在高维度上，SMC 算法可以有效地控制重要性采样权重的变化，并以小于 d 指数的成本估计固定维度边际，并表明重采样会导致蒙特卡罗误差减少并增加 ESS。我们所有的分析都是在目标密度为独立同分布的假设下进行的。
We investigate the stability of a Sequential Monte Carlo (SMC) method applied to the problem of sampling from a target distribution on 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 epsilon(N) as d -> infinity with 1 < epsilon(N) < N. The convergence is achieved with a computational cost proportional to Nd-2. If epsilon(N) infinity and lim(m ->infinity) epsilon(N,m) = N. Also, we show that the Monte Carlo error for estimating a fixed-dimensional marginal expectation is of order 1/root 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.