Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models
Unbiased Monte Carlo: Posterior estimation for intractable/infinite-dimensional models
复制标题
DOI：
10.3150/16-bej911
复制
发表时间：
2014-11
期刊：
影响因子：
1.5
通讯作者：
S. Agapiou;Gareth O. Roberts;Sebastian J. Vollmer
中科院分区：
文献类型：
--
作者：
S. Agapiou;Gareth O. Roberts;Sebastian J. Vollmer
我们给出了难解随机模型的无偏估计的一般方法。我们考虑目标分布可以被写成分布的适当极限的情况，以及传统方法需要截断这种表示导致系统偏差的情况。例如，目标分布可以表示为适当希尔伯特空间中的基展开的$L^2$-极限；或者，兴趣分布可以表示为随机变量序列的弱极限，如在MCMC中。我们的主要动机来自无限维模型，这些模型可以用基函数的一系列展开式来描述(例如由Karhunen-Love展开式给出的展开式)。我们考虑沿这种扩展的直接无偏估计方案，以及那些基于MCMC方案的方案，由于其维度的原因，不能直接实现，但可以有效地无偏估计。对于我们的所有方法，我们都给出了证明稳健蒙特卡罗实现的数值稳定性的理论，在某些情况下，我们用模拟来说明。有趣的是，我们的方法的计算效率通常与更简单的方法相当，后者是有偏见的。我们所提出的方法的有效性的关键是构造适当的耦合，其中许多耦合与过去算法及其变体的耦合中使用的蒙特卡罗结构强烈共鸣。
We provide a general methodology for unbiased estimation for intractable stochastic models. We consider situations where the target distribution can be written as an appropriate limit of distributions, and where conventional approaches require truncation of such a representation leading to a systematic bias. For example, the target distribution might be representable as the $L^2$-limit of a basis expansion in a suitable Hilbert space; or alternatively the distribution of interest might be representable as the weak limit of a sequence of random variables, as in MCMC. Our main motivation comes from infinite-dimensional models which can be parame- terised in terms of a series expansion of basis functions (such as that given by a Karhunen-Loeve expansion). We consider schemes for direct unbiased estimation along such an expansion, as well as those based on MCMC schemes which, due to their dimensionality, cannot be directly imple- mented, but which can be effectively estimated unbiasedly. For all our methods we give theory to justify the numerical stability for robust Monte Carlo implementation, and in some cases we illustrate using simulations. Interestingly the computational efficiency of our methods is usually comparable to simpler methods which are biased. Crucial to the effectiveness of our proposed methodology is the construction of appropriate couplings, many of which resonate strongly with the Monte Carlo constructions used in the coupling from the past algorithm and its variants.