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Markov Chain Monte Carlo in small worlds

机译：马尔可夫链蒙特卡洛小世界

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摘要

As the number of applications for Markov Chain Monte Carlo (MCMC) grows, the power of these methods as well as their shortcomings become more apparent. While MCMC yields an almost automatic way to sample a space according to some distribution, its implementations often fall short of this task as they may lead to chains which converge too slowly or get trapped within one mode of a multi-modal space. Moreover, it may be difficult to determine if a chain is only sampling a certain area of the space or if it has indeed reached stationarity. In this paper, we show how a simple modification of the proposal mechanism results in faster convergence of the chain and helps to circumvent the problems described above. This mechanism, which is based on an idea from the field of "small-world" networks, amounts to adding occasional "wild" proposals to any local proposal scheme. We demonstrate through both theory and extensive simulations, that these new proposal distributions can greatly outperform the traditional local proposals when it comes to exploring complex heterogenous spaces and multi-modal distributions. Our method can easily be applied to most, if not all, problems involving MCMC and unlike many other remedies which improve the performance of MCMC it preserves the simplicity of the underlying algorithm.
机译：随着马尔可夫链蒙特卡洛（MCMC）的应用程序数量的增加，这些方法的功能及其缺点变得更加明显。尽管MCMC产生了一种根据某种分布对空间进行采样的几乎自动的方法，但是它的实现常常达不到这一任务，因为它们可能导致链收敛太慢或陷入多模态空间的一种模式中。此外，可能很难确定一条链是仅对空间的某个区域进行采样，还是确实已达到平稳状态。在本文中，我们展示了对提议机制的简单修改如何导致链的更快收敛，并有助于规避上述问题。这种机制基于“小世界”网络领域的想法，相当于将偶然的“野生”建议添加到任何本地建议方案中。我们通过理论和广泛的模拟证明，在探索复杂的异构空间和多模式分布时，这些新的提议分布可以大大优于传统的本地提议。我们的方法可以轻松地应用于大多数（即使不是全部）涉及MCMC的问题，并且与许多其他改善MCMC性能的补救措施不同，它保留了底层算法的简单性。

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