在高維度統計問題上，當變數間具有複雜的相依關係，且不知道完整的聯合分配時，通常可藉由選定一些低維的條件分配來協助建立模型。因此，一個重要的問題是：對於選定的條件分配，是否存在共同的聯合分配？若存在，我們稱它們爲相容。接著，在相容的情況下，又該如何找出所有可能的聯合分配？這些問題激發出許多關於條件分配相容性議題的研究。
本文中聚焦於二維有限離散條件分配 X|Y 與 Y|X，我們提出了一套新的理論架構，除了能夠處理相容性議題外，也可應用到不同的研究領域上。利用選定的條件分配 X|Y 與 Y|X，Kuo and Wang (2019)建構出對應的轉移矩陣 T1 與 T2，以及轉移矩陣 T12 = T1T2 與 T21 = T2T1。在一般的條件分配模型（放寬支撐集相同的要求）下，運用已知的隨機過程理論，我們發展出一套有關T1、T2、T12 與 T21 的嶄新隨機理論，不僅可解決傳統的相容性議題（支撐集相同），也可解決廣義的相容性議題（支撐集可相異）。此外，利用這套理論，我們也對吉布斯抽樣（系統式掃描或隨機式掃描）的收斂行為進行分析，並將收斂行為的表現跟條件分配相容性的結果連結起來。同時，我們也提供了利用吉布斯抽樣建模的演算法。最後，我們也嘗試將這套理論應用在有限兩人非合作一般和賽局上，並提出了一個求均衡解的新概念與方法。

In high-dimensional statistical problems, when variables exhibit complex dependence structures and the full joint distribution is unknown, it is often advantageous to specify a collection of low-dimensional conditional distributions to facilitate model construction. This naturally gives rise to a fundamental question: given a set of conditional distributions, does there exist a joint distribution that is consistent with them? If such a joint distribution exists, the conditional distributions are said to be `compatible'. Furthermore, when compatibility holds, how can all possible compatible joint distributions be characterized? These questions have motivated extensive research on the compatibility of conditional distributions.
This dissertation focuses on two-dimensional finite discrete conditional distributions, X | Y and Y | X. We develop a new theoretical framework that not only addresses compatibility issues but also extends to applications in other research areas. Based on the specified conditional distributions X | Y and Y | X, Kuo and Wang (2019) constructed the corresponding transition matrices T1 and T2, together with the composite transition matrices T12 = T1T2 and T21 = T2T1. Under general conditional distribution models, where the requirement of identical support sets is relaxed, we develop a novel stochastic theory for T1, T2, T12, and T21 by employing established results from stochastic process theory. This framework not only resolves the classical compatibility issues (where the support sets are identical) but also addresses the generalized compatibility issues (where the support sets may differ). Moreover, using this framework, we analyze the convergence behavior of Gibbs sampling under both systematic-scan and random-scan schemes, and establish a connection between its convergence properties and the compatibility of the specified conditional distributions. We also provide two algorithms for model construction based on Gibbs sampling. Finally, we explore the application of this theoretical framework to finite two-person non-cooperative general-sum games and introduce a new concept and method for computing equilibrium solutions.

顧仲航. (2011). 以特徵向量法解條件分配相容性問題 . (未出版之碩士論文). 國立政治大學應用數學系，臺北市.
Arnold, B. C., Castillo, E., & Sarabia, J. M. (2002). Exact and near compatibility of discrete conditional distributions. Computational Statistics & Data Analysis, 40(2), 231–252.
Arnold, B. C., Castillo, E., & Sarabia, J. M. (2004). Compatibility of partial or complete conditional probability specifications. Journal of Statistical Planning and Inference, 123(1), 133–159.
Arnold, B. C., & Press, S. J. (1989). Compatible conditional distributions. Journal of the American Statistical Association, 84(405), 152–156.
Betten, A., Braun, M., Fripertinger, H., Kerber, A., Kohnert, A., & Wassermann, A. (2006). Error-correcting linear codes: Classification by isometry and applications. Springer Berlin Heidelberg.
Chang, S.-H., Kuo, K.-L., Song, C.-C., & Jiang, T. J. (2026+). Convergence of systematic-scan and random-scan Gibbs samplers for bivariate discrete conditional distributions. Annals of the Institute of Statistical Mathematics. In Press.
Chen, S.-H., & Ip, E. H. (2015). Behaviour of the Gibbs sampler when conditional distributions are potentially incompatible. Journal of Statistical Computation and Simulation, 85(16), 3266–3275.
Dixit, A. K., Skeath, S., & McAdams, D. (2020). Games of strategy. W.W. Norton & Company.
Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, PAMI-6(6), 721–741.
Heckerman, D., Chickering, D. M., Meek, C., Rounthwaite, R., & Kadie, C. (2000). Dependency networks for inference, collaborative filtering, and data visualization. Journal of Machine Learning Research, 1, 49–75.
Hoel, P. G., Port, S. C., & Stone, C. J. (1972). Introduction to stochastic processes. Houghton Mifflin Company.
Hughes, R. A., White, I. R., Seaman, S. R., Carpenter, J. R., Tilling, K., & Sterne, J. A. C. (2014). Joint modelling rationale for chained equations. BMC Medical Research Methodology, 14, Article 28. Retrieved from https://doi.org 10.1186/1471-2288-14-28
Kuo, K.-L., Song, C.-C., & Jiang, T. J. (2017). Exactly and almost compatible joint distributions for high-dimensional discrete conditional distributions. Journal of Multivariate Analysis, 157, 115–123.
Kuo, K.-L., & Wang, Y. J. (2019). Pseudo-Gibbs sampler for discrete conditional distributions. Annals of the Institute of Statistical Mathematics, 71, 93–105.
Levine, R. A., & Casella, G. (2006). Optimizing random scan Gibbs samplers. Journal of Multivariate Analysis, 97(10), 2071–2100.
Liu, J. S., Wong, W. H., & Kong, A. (1995). Covariance structure and convergence rate of the Gibbs sampler with various scans. Journal of the Royal Statistical Society. Series B, 57(1), 157–169.
McCain, R. A. (2014). Reframing economics: Economic action as imperfect cooperation. Edward Elgar Publishing.
Meyer, C. D. (2000). Matrix analysis and applied linear algebra. SIAM.
Moulin, H. (1986). Game theory for the social sciences. NYU press.
Nash, J. (1951). Non-cooperative games. Annals of Mathematics, 54(2), 286–295.
Owen, G. (2013). Game theory. Emerald Group Publishing Limited.
Peterson, M. (2015). The prisoner’s dilemma. Cambridge University Press.
Rodrigues, A., Koeppl, H., Ohtsuki, H., & Satake, A. (2009). A game theoretical model of deforestation in human–environment relationships. Journal of Theoretical Biology, 258(1), 127–134.
Smith, A. (1776). The wealth of nations. W. Strahan and T. Cadell, London.
Song, C.-C., Li, L.-A., Chen, C.-H., Jiang, T. J., & Kuo, K.-L. (2010). Compatibility of finite discrete conditional distributions. Statistica Sinica, 20(1), 423–440.
Van Buuren, S., Brand, J. P. L., Groothuis-Oudshoorn, C. G. M., & Rubin, D. B. (2006). Fully conditional specification in multivariate imputation. Journal of Statistical Computation and Simulation, 76(12), 1049–1064.
Yates, R. D., & Goodman, D. J. (2005). Probability and stochastic processes: A friendly introduction for electrical and computer engineers. John Wiley & Sons, Inc.