實務上，我們常需要建構母體的機率模型，然而當母體所涉及的隨機變數愈多，亦即維度愈高時，直接建構高維度之聯合分配的難度就越高，故我們可試著透過一組維度較低的條件分配來獲得聯合分配，而是否存在聯合分配滿足這一組條件分配，即為所謂的相容性問題。

本文首先將二維條件分配的相容性問題跟馬可夫鏈的對應關係做詳細比較。我們發現，由於二維條件分配所對應的馬可夫鏈是非週期性的，因此，Arnold(1989)利用馬可夫鏈的理論，提出聯合分配唯一存在的充要條件，可做進一步化簡。

給定二維條件分配，若不能找到共同的聯合分配則稱他們是不相容的；不相容的程度有各種衡量指標，而這些指標之間的關係也是值得我們研究的課題。在條件分配不相容情況下，我們先透過模擬數據的方式，對Arnold, Castillo, Sarabia (2002)及顧仲航 (2011)所提出的四種不相容測度值進行觀察，試圖進一步獲得有關他們之間關係的資訊。接著，在2x2的條件機率矩陣下，我們推導出四種不相容測度ϵ1、ϵ2、ϵ3、ϵ4以及對偶測度ϵ3*的計算公式，而且獲得2ϵ1=ϵ2=ϵ3=ϵ3*=ϵ4 的結果。最後，在2xJ的條件機率矩陣下，J>=2，我們推導出ϵ2 與ϵ3的計算公式，並且證明出ϵ2=ϵ3的關係；同時也在Ix2的條件機率矩陣下，I>=2，推出ϵ2=ϵ3*。

In practice, we may need to construct a joint distribution for a population. However, when the dimension of the random variables corresponding to the population is higher, it is often more difficult to find such a high dimensional joint distribution. Hence, we can obtain a set of lower dimensional conditional distributions first, and then use them to find their corresponding joint distribution. If there is a joint distribution matching this set of conditional distributions, we say this set of conditional distributions is compatible.

First, we study the relationship between the compatibility and Markov chain. Since the Markov chain corresponding to two dimensional conditional distributions is aperiodic, we can further simplfy the necessary and sufficient condition of uniquness of a joint distribution given by Arnold(1989).

The two dimensional conditional distributions are called incompatible if there is no common joint distribution for them. There are a few measures of degree of incompatibility in literature. Our aim is to study, through the simulations first, the relation among the four measures of degree of incompatibility given by Arnold, Castillo, Sarabia (2002) and Ku (2010). We derive the computational formulas for these four measures of degree of incompatibility ϵ1, ϵ2, ϵ3, ϵ4 and the duality measure ϵ3* under 2x2 conditional probability matrices and prove that 2ϵ1=ϵ2=ϵ3=ϵ3*=ϵ4. In addition, we derive the computational formulas for ϵ2 and ϵ3 and prove that ϵ2=ϵ3 under 2xJ conditional probability matrices, where J>=2. Finally, we also show that ϵ2=ϵ3* under Ix2 conditional probability matrices, where I>=2.

參考文獻
Arnold, B. C., Castillo, E., and Sarabia, J. M. (2002). Exact and near compatibility of discrete conditional distributions. Comput. Stat. Data Anal., 40(2):231–252.
Arnold, B. C. and Press, S. J. (1989). Compatible conditional distributions. Journal of the American Statistical Association, 84(405):152–156.
Song, C.-C., Li, L.-A., Chen, C.-H., Jiang, T. J., and Kuo, K.-L. (2010). Compatibility of finite discrete conditional distributions. Statistica Sinica 20 (2010).
Yates, R. and Goodman, D. (2005). Probability and Stochastic Processes: A Friendly Introduction for Electrical and Computer Engineers. John Wiley & Sons, second edition.
黃文璋(1995). 隨機過程. 華泰文化事業股份有限公司.
顧仲航(2011). 以特徵向量法解條件分配相容性問題. 國立政治大學應用數學系碩士論文.