二維指數分配之貝氏分析

Bayesian Analysis of Moran–Downton Bivariate Exponential Distribution

賴駿騰(Jun-Teng Lai)

指導教授 ： 林余昭

中原大學/理學院/應用數學研究所/碩士(2018年)

https://doi.org/10.6840/cycu201800685

摘要

貝氏統計分析已廣泛應用在醫學、教育、工業等領域，雖然以往受限於計算較為複雜，但隨著科技的進步，許多相關統計軟體的出現，統計學者可以利用過去的經驗，再結合蒐集的資訊進行分析與模擬，然而這些問題得到了解答，使得貝氏統計越來越受到重視。利用 OpenBUGS 軟體可以有效的執行馬可夫鏈蒙地卡羅法來估計模型的參數。本文要討論 Moran-Downton 二維指數分配參數的貝氏估計過程，當正確的使用已知的訊息，再對參數進行迭代後，再觀察參數的估計值是否逼近參數的真實值。所以本文以 R 軟體的 R2OpenBUGS 套件來對 Moran-Downton 二維指數分配的模型進行參數的貝氏統計分析。在 R 的環境下處理資料較為方便，再利用 R2OpenBUGS 套件執行貝氏統計的參數估計。

關鍵字

貝氏統計 ； 馬可夫鏈蒙地卡羅法 ； R2OpenBUGS 套件

並列摘要

The Bayesian analysis has many applications in almost all fields, such as education, medical treatments, quality control, and so on. But the computation of Bayesian statistics is more complicated compared to the
frequentists statistics since the statisticians usually need to write the programs codes to apply MCMC method.
In this thesis, we consider the Bayesian approach of Moran-Downtown bivariate Exponential (DBE) model and analyze the data using OpenBUGS. The freeware OpenBUGS is a popular Bayesian software with a simple language
syntax, but its interface is tedious in data analysis and simulation study.
We first simulate the bivariate data from the model and use the R package,R2OpenBUGS, to call OpenBUGS from R console to analyze both the simulated data and real data. The difficulty of this research is that the probability density of DBE model consisting of modified Bessel function is not standard in the BUGS language. We successfully use the Zero trick to describe the DBE alternative likelihood in terms of a sum of geometric and gamma functions.

並列關鍵字

Bayesian statistics ； Markov chain Monte Carlo method ； R2OpenBUGS package

參考文獻

Hastings, W. K. 1970. Monte Carlo Sampling Methods Using Markov Chains andTheir Applications, Biometrika, 57, 97-109.

Metropolis, N., Rosenbluth, A. W., Rosenbluth,M. N. , Teller,A. H., and Teller, E.1953. Equations of State Calculations by Fast Compution Machines, Journal of

chemical Physics, 21, 1087-1091.

Geman, S. and Geman, D. (1984). Stochastic Relaxation, Gibbs Distribution and the Bayesian Restoration of Images. IEEE Trans. Patten Ann., Machine Intell.,6,721-741

Gelfand,A. E. and Smith, A. F. M. (1990).Sampling-Based Approaches to Calculating Marginal Densities. J. Am. Statist. Assoc., 85,398-409.