加權蒙地卡羅模擬之進階研究

Abstract

因為電腦的計算速度愈來愈快，因此有許多科學上的問題愈來愈需要仰賴數值計算。 目前工程學家或是生物資訊學家遇到的問題都非常複雜，常常有高達 上百維度的積分 或其他問題，傳統的數值方法皆無法解決，在最近三十年內蒙地卡羅方法逐漸受到科 學家們的重視。眾所皆知，當我們在使用蒙地卡羅模擬方法時，一定要滿足 detail balance。如果不滿足，馬可夫鏈將不會收斂到要求之分配。因此，Metropolis 演算 法扮演非常重要的角色，Metropolis 演算法最大的缺點是常常消耗掉大量的電腦資 源。因此統計學家提出各種可能方法，加權蒙地卡羅演算法為其中很重要的一種，此 方法很容易跳出局部極值，以最快速度達到 ergodic。但為了達到此目的，我們也必 須付出一些代價，必須犧牲 detail balance 的性質，取而代之的是加重此一局部極值 的權重。通常此權重含有未知參數得函數，因此無法得知，我們必須估計他或想其他 辦法代替。因此，本計畫之主要目的在於探討加權蒙地卡羅方法的理論基礎。我們將 分成幾個主要步驟進行。第一：我們將探討在有限的馬可夫鏈時，加權蒙地卡羅方法 所產成的權重到底是具有何種分配，特別是在於其尾端的分配，即權重之尾端機率分 配為何? 這個分配將對權重的選取與如何利用權重做計算有極重要之影響。因此，我 們希望此計畫能解決此問題。如果可能，我們還希望能更進一步討論離散且具無窮點 時的馬可夫鏈，看看是否與有限點時具有一樣之權重尾端分配。

In the statistical simulation filed, the Metropolis algorithm is one of the most important ideals. The Metropolis algorithm has two steps. The first step is a random selection; the second step is to decide the acceptance probability. It is well know the Metropolis algorithm converges to stationary distribution very slow. It is very often to stay in a local maximum or minimum. Therefore many improvements have been proposed. Among those improvements, the dynamic weighting method introduced by Wong and Liang is one of the most important MCMC methods. The purpose of putting weights into the Monte Carlo process is to make the chain easily to move out from local maximum or minimum. Unfortunately, the dynamic weighting MCMC method does not satisfy the detail balance property which is the most important property to insure that the chain will converge to the target distribution. Instead of detail balance property, Wong and Liang proposed the IWIW property. In this project, we will focus on the property of IWIW for finite state Markov chain. In this case, we hope that we can find the limiting distribution of the weight function. Also, we will discuss those dynamic weighting method to see why they work well even the weighting functions have heavy tail probability. Another important subject of this project is to extend those results in finite state weighted Markov chain to infinite discrete weighted Markov chain. We hope that we can find the exact behavior of the tail probability in those discrete weighted Markov chain.