THFan

研究方向

貝氏分析最常被詬病的是其使用主觀性的先驗分佈，然而貝氏分析對估量的優劣評估相對地較其他方法簡單。因此客觀的貝氏方法是必要且重要的，而使用無資訊先驗分佈的貝氏分析通常可得到令貝氏學派和頻率學派都滿意的結論。

很多二次反應曲線模型都特別對反應變數之極值和產生極值之解釋變數的位置感興趣。我們發現一般最常用的均勻無資訊先驗分佈無法適用在我們考慮的直接模型的參數估計上，而另一種也很常用的 Jeffreys’先驗分佈卻會導致估計誤差被嚴重低估的現象，蓋因參數個數太多之故。我們使用參考先驗分佈將有興趣的參數和干擾參數分開，則可以得到比較合理的結果。

貝氏假設檢定方法可以應用到模型選擇的問題上，並且可以提供比古典的檢定(選模)方法更好的結果。然而在單點虛無假設上一般貝氏方法卻不能用不可積分的無資訊先驗分佈。而在遍歷且具有限狀態空間之馬可夫鏈模型中，由於參數空間是有界的，因此我們使用均勻和Jeffreys’先驗分佈來對馬可夫鏈之次數作模型選擇。我們的結果較一般常用的AIC和BIC方法在小樣本時要好很多，而大樣本時則沒有太大的差別。

我們也考慮以潛在貝氏因子來偵測在無界參數空間下改變點的個數。類似的方法也將應用到統計模型非常複雜的隱藏馬可夫鏈之秩的選模上。所有這些問題都會牽涉到複雜的計算問題，而MCMC和EM方法將用來解決這些統計計算的困難。

我們並以經驗貝氏的方法來分析台灣花蓮地區之地震資料。台灣地區地震發生頻繁，故資料很多，氣象局有很完整的資料。早期的資料可以用來提供先驗分佈的訊息，因此經驗貝氏是很合理的方法，MCMC方法也降低了計算的難度，我們的結果不遜於古典的分析法。

Noninformative Bayesian Analysis and Bayesian Model Selections

Bayesian analysis is often criticized for the use of subjective priors, but Baysian analysis is relatively easier in assessing the behavior of the estimators. An objective solution is thus important and necessary. Bayesian analysis with noninformative priors is usually satisfactory from both the Bayesian and the frequentist perspectives.

Many applications of a quadratic response model are specifically focused on estimation for the optimum response, maximum or minimum, and the point in the factor space where the optimum occurs. We discovered that putting the uniform priors on the parameters of interest directly is not suitable for the model considered, and the Jeffreys’ priors yield considerably underestimated estimation errors in the presence of too many parameters. The reference priors, derived by dividing the parameter vector into parameters of interest and nuisance parameters, however gare more reasonable results.

Bayesian hypothesis testing procedures can be applied to the problem of model selections which usually provide more appealing results than classical approaches. However, the standard Bayesian approach is not possible via improper noninformative priors for point null hypotheses. In the case of order determination of an ergodic finite state Markov chain, the parameter space is bounded, we therefore can use the uniform as well as the Jeffreys’ priors to select the correct order. Our results are more convincing than those obtained by the commonly used AIC and BIC approaches, at least in small samples.

We have also considered the intrinsic Bayes factor approach to detect the number of change points in problems in which the parameter spaces are unbounded. A similar idea is applied to determine the order of a hidden Markov chain which has a more complex statistical model. All problems discussed meet certain computational difficulties which are undertaken by the MCMC approaches as well as the EM algorithms.

Selected Recent Publications:

1. Fan, T. H. (1998). Noninformative Bayesian estimation for the optimum in a single factor quadratic response model. (Submitted.)

2. Fan, T. H. and Tsai, C. A. (1999). A Bayesian method in determining the order of a finite state Makov chain. To appear in Communications in Statistics：Theory and Method.

3. Fan, T. H. and Tsai, C. A. (1998). Intrinsic Bayes factors in detecting the number of change points. (Manuscript).