论文标题

马尔可夫链蒙特卡洛法的有效介绍

An effective introduction to the Markov Chain Monte Carlo method

论文作者

Wang, Wenlong

论文摘要

我们使用简单的人口动态模型作为我们的动机,并专注于一些基本分布,对著名的马尔特·卡洛方法进行直观,概念但半符合的介绍。从概念上讲,城市之间的人口流与在状态空间中的单个步行者的随机步行非常相似。我们从两个州(然后是三个州)开始,最后将设置完全概括为离散和连续分布的许多州。尽管具有数学的简单性,但该设置明显地包括马尔可夫链蒙特卡洛的所有基本概念,而不会丧失一般性,例如,千古,全球平衡和详细的平衡,提案或选择概率,接受概率,最高为潜在的随机矩阵和错误分析。我们的教学经验表明,大多数物理学的高级本科生都可以毫无困难地遵循这些材料。

We present an intuitive, conceptual, but semi-rigorous introduction to the celebrated Markov Chain Monte Carlo method using a simple model of population dynamics as our motivation and focusing on a few elementary distributions. Conceptually, the population flow between cities closely resembles the random walk of a single walker in a state space. We start from two states, then three states, and finally the setup is fully generalized to many states of both discrete and continuous distributions. Despite the mathematical simplicity, the setup remarkably includes all the essential concepts of Markov Chain Monte Carlo without loss of generality, e.g., ergodicity, global balance and detailed balance, proposal or selection probability, acceptance probability, up to the underlying stochastic matrix, and error analysis. Our teaching experience suggests that most senior undergraduate students in physics can closely follow these materials without much difficulty.

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