A lognormal central limit theorem for particle approximations of normalizing constants
A lognormal central limit theorem for particle approximations of normalizing constants
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DOI：
10.1214/ejp.v19-3428
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发表时间：
2014-10-07
影响因子：
1.4
通讯作者：
Doucet, Arnaud
中科院分区：
文献类型：
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作者：
Berard, Jean;Del Moral, Pierre;Doucet, Arnaud
费曼-卡克模型出现在包括物理、化学和信号处理在内的许多科学学科中。它们的平均场粒子解释，通常被称为顺序蒙特卡罗或粒子过滤器，因为它们允许从复杂的概率分布序列中进行近似采样并估计其相关的归一化常数，因此得到了广泛的应用。众所周知，在正则性假设下，这些归一化常数估计的相对方差随着时间范围n线性增加，因此实践者通常线性地缩放粒子数N，从而获得其相对方差保持一致有界的估计。我们在这里建立，在这个标准的线性标度策略下，归一化常数估计的波动是对数正态的，因为n，因此，N趋于无穷大。对于时齐环境中的粒子吸收模型和遍历随机环境中的隐马尔可夫模型，我们还给出了极限偏差和方差的更明确的描述。
Feynman-Kac models arise in a large variety of scientific disciplines including physics, chemistry and signal processing. Their mean field particle interpretations, termed commonly Sequential Monte Carlo or Particle Filters, have found numerous applications as they allow to sample approximately from sequences of complex probability distributions and estimate their associated normalizing constants. It is well-known that, under regularity assumptions, the relative variance of these normalizing constant estimates increases linearly with the time horizon n so that practitioners usually scale the number of particles N linearly w.r.t n to obtain estimates whose relative variance remains uniformly bounded w.r.t n. We establish here that, under this standard linear scaling strategy, the fluctuations of the normalizing constant estimates are lognormal as n, hence N, goes to infinity. For particle absorption models in a time-homogeneous environment and hidden Markov models in an ergodic random environment, we also provide more explicit descriptions of the limiting bias and variance.