Chen(2010)提出如何用勝率函數來判斷給定的連續條件分配是否相容，以及 相容時如何求對應的聯合分配。本研究提出，在二維有限的情形下，如何用勝率 矩陣來判斷給定的條件機率矩陣是否相容，以及相容時如何求對應的聯合機率矩 陣。又給定的條件機率矩陣不相容時，我們介紹了四種修改勝率矩陣的方法，同 時在使用幾何平均法調整勝率矩陣的過程中，也發現選取最佳參考點以獲得最佳 近似聯合機率矩陣之方法，並且給予理論證明。最後以模擬的方式發現，在修改 勝率矩陣的四種方法中，以幾何平均法所得到的近似聯合機率矩陣，其條件機率 矩陣最常接近所給定的條件機率矩陣。 / Chen (2010) provides the representations of odds ratio function to examine the compatibility of conditional probability density functions and gives the corresponding joint probability density functions if they are compatible. In this research, we provide the representations of odds ratio matrix to examine the compatibility of two discrete conditional probability matrices and give the corresponding joint probability matrix if they are compatible. For incompatible situations, we offer four methods to revise odds ratio matrices to find near joint probability matrices so that their conditional probability matrices are not far from the two given ones. That is, we provide four methods so that the sums of error squares are small. For each method, the sum of error squares may depend on the same reference point of two odds ratio matrices. We first discover by example that only the geometric method out of these four methods has a pattern to get the best reference point so that the sum of error squares is smallest. We then prove this finding in general. In addition, through simulation results, the geometric method would provide the smallest sum of error squares most often among these four methods. Hence, we suggest using geometric method. Its strategy to find the best reference point is also given.

給定兩個條件機率分配，判斷他們是否相容？是否有唯一的聯合機率分配？以及相容時，如何找出所有可能的聯合機率分配？是研究相容性相當重要的課題。本文針對有限離散型二維條件機率分配，以Arnold and Press(1989) 最先提出的比值矩陣法，及由Song , Li, Chen, Jiang, and Kuo (2010) 所提出的檢驗法為架構，提出新演算法且利用此演算法來設計程式，使程式能判斷兩條件機率分配是否相容，以及相容後可求出對應的所有聯合機率分配。本文亦依據新演算法並應用MATLAB軟體設計程式，讓使用者可以很快地對上述三個問題得到答案。 / When two conditional distributions are given, the following three important questions are likely to be raised. Are they compatible? Is the corresponding joint distribution unique if they are compatible? How do you find all the corresponding joint distributions if they are compatible? In this thesis, basing on ratio matrix method given first by Arnold and Press (1989), and on the method for checking compatibility existence, for checking uniqueness, and for finding all possible joint distributions provided by Song, Li, Chen, Jiang, and Kuo (2010), we provide a new algorithm to answer these questions. Using this new algorithm, we also provide a MATLAB computer program so that any user could get the answer quickly for the above three questions.

中文摘要 有限離散條件分配相容性問題可依相容性檢驗、唯一性檢驗以及找出所有的聯合機率分配三層次來討論。目前的文獻資料有幾種研究方法，本文僅分析、比較其中的比值矩陣法和圖形法。 二維中，我們發現簡化二分圖的分支與IBD矩陣中的對角塊狀矩陣有密切的對應關係。在檢驗相容性時，圖形法只需檢驗簡化二分圖中的每個分支，正如同比值矩陣法只需檢驗IBD矩陣中的每一個對角塊狀矩陣即可。在檢驗唯一性時，圖形法只需檢驗簡化二分圖中的分支數是否唯一，正如同比值矩陣法只需檢驗IBD矩陣中的對角塊狀數是否唯一即可。在求所有的聯合機率分配時，運用比值矩陣法可推算出所有的聯合機率分配，但是圖形法則無法求出。 三維中，本文提出了修正比值矩陣法，將比值數組按照某種索引方式在平面上有規則地呈現，可降低所需處理矩陣的大小。此外，我們也發現修正比值矩陣中的橫直縱迴路和簡化二分圖中的迴路有對應的關係，因此可觀察出兩種方法所獲致某些結論的關聯性。在檢驗唯一性時，圖形法是檢驗簡化二分圖中的分支數是否唯一，而修正比值矩陣法是檢驗兩個修正比值矩陣是否分別有唯一的GROPE矩陣。修正比值矩陣法可推算出所有的聯合機率分配。 圖形法可用於任何維度中，修正比值矩陣法也可推廣到任何維度中，但在應用上，修正比值矩陣法比圖形法較為可行。 / The issue of the compatibility of finite discrete conditional distributions could be discussed hierarchically according to the compatibility, the uniqueness, and finding all possible joint probability distributions. There are several published methods, but only the Ratio Matrix Method and the Graphical Method are analyzed and compared in this thesis. In bivariate case, a close correspondence between the components of the reduced bipartite graph and the diagonal block matrices of the IBD matrix can be found. When we examine the compatibility, just as simply each diagonal block matrix of the IBD matrix needs to be examined using the Ratio Matrix Method, so does each component of the reduced bipartite graph using the Graphical Method. When we examine the uniqueness, just as whether the number of the diagonal blocks of the IBD matrix is unique needs to be examined, so does the number of the components of the reduced bipartite graph. The Ratio Matrix Method can provide all possible joint probability distributions, but the Graphical Method cannot. In trivariate case, this thesis proposes a Revised Ratio Matrix Method, in which we can present the ratio array regularly in the plane according to the index and reduce the corresponding matrix size. It is also found that each circuit in the revised ratio matrix corresponds to a circuit in the reduced bipartite graph. Therefore, the relation between the results of the two methods can be observed. When we examine the uniqueness with the Graphical Method, we examine whether the number of the components in the reduced bipartite graph is unique. But with the Revised Ratio Matrix Method, we examine whether each revised ratio matrix has a unique GROPE matrix. All possible joint probability distributions can be derived through the Revised Ratio Matrix Method. The Graphical Method can be applied to the higher dimensional cases, so can the Revised Ratio Matrix Method. But the Revised Ratio Matrix Method is more feasible than the Graphical Method in application.

給定兩個有限離散型條件分配，我們可以去探討有關相容性及唯一性的問題。Tian et al.(2009)提出一個統合的方法，將相容性的問題轉換成具限制條件的線性方程系統(以邊際機率為未知數)，並藉由 l_2-距離測量解之誤差，進而求出最佳解來。他們也提出了電腦數值計算法在檢驗相容性及唯一性時的準則。 由於 Tian et al.(2009)的方法是把邊際機率和為 1 的條件放置在線性方程系統中，從理論的觀點來看，我們認為該條件在此種做法下未必會滿足。因此，本文中將邊際機率和為 1 的條件從線性方程系統中抽離出來，放入限制條件中，再對修正後的問題求最佳解。 我們提出了兩個解決問題的方法：(一) LRG 法；(二) 干擾參數法。LRG 法是先不管機率值在 0 與 1 之間的限制，在邊際機率和為 1 的條件下，利用 Lagrange 乘數法導出解的公式，之後再利用 Rao-Ghangurde 法進行修正，使解滿足機率值在 0 與 1 之間的要求。干擾參數法是在 Lagrange 乘數法公式解中有關廣義逆矩陣的計算部份引進了微量干擾值，使近似的逆矩陣及解可快速求得。理論證明，引進干擾參數所增加的誤差不超過所選定的干擾值，易言之，由干擾參數法所求出的解幾近最佳解。故干擾參數法在處理相容性問題上，是非常實用、有效的方法。從進一步分析Lagrange 乘數法公式解的過程中，我們也發現了檢驗條件分配"理論"相容的充分條件。 最後，為了驗證 LRG 法與干擾參數法的可行性，我們利用 MATLAB 設計了程式來處理求解過程中的運算，並以 Tian et al.(2009)文中四個可涵蓋各種情況的範例來解釋說明處理的流程，同時將所獲得的結果和 Tian et al. 的結果做比較。 / Given two finite discrete conditional distributions, we could study the compatibility and uniqueness issues. Tian et al.(2009) proposed a unified method by converting the compatibility problem into a system of linear equations with constraints, in which marginal probability values are assumed unknown. It locates the optimum solution by means of the error of l_2 - discrepancy. They also provided criteria for determining the compatibility and uniqueness. Because the condition of sum of the marginal probability values being equal to one is in Tian et al.s’linear system, it might not be fulfilled by the optimum solution. By separating this condition from the linear system and adding into constraints, we would look for the optimum solution after modification. We propose two new methods: (1) LRG method and (2) Perturbation method. LRG method ignores the requirement of the probability values being between zero and one initially, it then uses the Lagrange multipliers method to derive the solution for a quadratic optimization problem subject to the sum of the marginal probability values being equal to 1. Afterward we use the Rao-Ghangurde method to modify the computed value to meet the requirement. The perturbation method introduces tiny perturbation parameter in finding the generalized inverse for the optimum solution obtained by the Lagrange multipliers method. It can be shown that the increased error is less than the perturbation value introduced. Thus it is a practical and effective method in dealing with compatibility issues. We also find some sufficient conditions for checking the compatibility of conditional distributions from further analysis on the solution given by Lagrange multipliers method. To show the feasibilities of LRG method and Perturbation method, we use MATLAB to device a program to conduct them. Several numerical examples raised by Tian et al.(2009) in their article are applied to illustrate our methods. Some comparisons with their method are also presented.