A framework for adaptive MCMC targeting multimodal distributions
A framework for adaptive MCMC targeting multimodal distributions
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DOI：
10.1214/19-aos1916
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发表时间：
2018-12
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影响因子：
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通讯作者：
E. Pompe;C. Holmes;K. Latuszy'nski
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文献类型：
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作者：
E. Pompe;C. Holmes;K. Latuszy'nski
本文提出了一种新的多峰分布抽样的蒙特卡罗方法。这种技术的思想是基于将任务分为两部分：找到目标分布$\pi$的模式，并在已知模式位置的情况下采样。采样算法依赖于两种类型的步骤：局部步骤，保留模式;以及跳转到与不同模式相关联的区域。此外，该方法在运行时学习算法的最佳参数，而不需要用户干预。我们的技术应该被认为是一个灵活的框架，其中的动作设计可以遵循各种策略，从广泛的MCMC文献。为了设计一个自适应方案，促进本地和跳跃移动，我们引入了一个辅助变量表示每个模式，我们定义了一个新的目标分布$\tilde{\pi}$上的增广状态空间$\mathcal{X}~\times~\mathcal{I}$，其中$\mathcal{X}$是原始状态空间$\pi$和$\mathcal{I}$是模式的集合。当算法运行并更新其参数时，目标分布$\tilde{\pi}$也会不断修改。这激发了一类新的算法，辅助变量自适应MCMC。我们证明了一般遍历的结果，为整个类之前专门的情况下，我们的算法。
We propose a new Monte Carlo method for sampling from multimodal distributions. The idea of this technique is based on splitting the task into two: finding the modes of a target distribution $\pi$ and sampling, given the knowledge of the locations of the modes. The sampling algorithm relies on steps of two types: local ones, preserving the mode; and jumps to regions associated with different modes. Besides, the method learns the optimal parameters of the algorithm while it runs, without requiring user intervention. Our technique should be considered as a flexible framework, in which the design of moves can follow various strategies known from the broad MCMC literature. In order to design an adaptive scheme that facilitates both local and jump moves, we introduce an auxiliary variable representing each mode and we define a new target distribution $\tilde{\pi}$ on an augmented state space $\mathcal{X}~\times~\mathcal{I}$, where $\mathcal{X}$ is the original state space of $\pi$ and $\mathcal{I}$ is the set of the modes. As the algorithm runs and updates its parameters, the target distribution $\tilde{\pi}$ also keeps being modified. This motivates a new class of algorithms, Auxiliary Variable Adaptive MCMC. We prove general ergodic results for the whole class before specialising to the case of our algorithm.