運用Dirichlet過程估計卜瓦松均數

Abstract

以往利用貝氏方法估計卜瓦松均數，為了計算的可行性，大多用伽碼分配（卜瓦松的共軛分配）當成均數的先驗分配，且先驗分配以經驗貝氏法來估計（母數經驗貝氏法），然而在卜瓦松均數背離伽碼分配的狀況下，估計效果並不佳。Laird (1978)提出無母數最大概似先驗分配估計法，提供卜瓦松均數之先驗分配另一選擇。當均數不具伽碼分配而集中在某些值時，此法有很好的估計效果;但在均數分散（變異數大）的狀況下，估計效果並不理想。由於在大多數的情況下，我們無法確定均數分配的型式，因此無從判定用何種估計方法較為妥當。本文首先嘗試用Escobar (1994)所提出的Dirichlet過程估計法來估計卜瓦松均數，並由模擬結果得知，不論均數之型態為伽碼分配或少數幾個值的離散分配，Dirichlet過程估計法的效果總是介於無母數最大概似估計法及母數經驗貝氏法之間，並趨向其中較好的估計法。

In the past, when using the Bayesian method to estimate Poisson means, we used to choose conjugate prior distribution for computational simplicity, and we also empirically estimated the prior of the means Gamma distribution (PEB). However, if the true distribution of the means departs from Gamma distribution, PEB method is not very efficient. Laird (1978) estimated the prior distribution by nonparametric maximum likelihood (NPML), which provided another choice of the prior distribution. When the means are clustered in few values instead of having Gamma distribution, NPML method is very efficient, but when the means are very disperse, the method is not efficient. Because, most of the time, we do not know the true distribution of the means, it is hard to decide whether to use PEB or NPML method. This research first try to estimate Poisson means by Dirichlet Process (DP) method which is developed by Escobar (1994). According to our simulation study, whether the distribution of the means is Gamma distribution or discrete distribution having few values, DP method is as good as PEB method when PEB method is better than NPML method, and it is as good as NPML method when NPML method is better.