繃散分析在不遺漏資訊的情況下，針對使用者的問題可提供一個可靠性高的參考決策；相對於繃散分析，整合分析只收集整合資料去估計問題的近似值。雖然收集整合資料所需的成本較繃散資料低，但整合分析尚必須付出整合誤差的錯誤成本，若貝氏分析具有完全整合的性質，則整合分析與繃散分析會具有相同的分析結果，因此，完全整合的性質實為降低分析成本的關鍵之一。實務上，定義在「非負」與「總和為一」的單位體之貝氏分析，常以狄氏分配為先驗分配，因為狄氏分配在計算上較為簡單，且具有完全整合的性質。然而，狄氏分配受限於任兩變數必須為負相關的限制，當參數向量存有顯著正相關的因子時，狄氏分配便不適合當成先驗分配。IBB 分配是透過隨機二元分枝過程所產生的多項式分配，且狄氏分配是IBB 分配的一個特例，以IBB 分配為先驗分配的貝氏分析，在計算上較狄氏分配複雜許多，但IBB 分配不限制參數向量任兩變數的正負相關性，此特性讓定義在單位體的貝氏分析在使用上更具彈性。本研究已證實出，IBB 分配在特定的條件之下具有完全整合的性質，並且提出在變數向量的期望值與共變數矩陣已知的情況下IBB 分配的架構方式。

Collecting data for a sufficient statistic is generally much easier and less expensive than
recording the details of the available data. When the posterior distributions of a quantity of interest given the aggregate and disaggregate data are identical, perfect aggregation is said to hold, and in this case the aggregate data is a sufficient statistic for the quantity of interest. The Dirichlet distribution is one of the most popular multivariate distributions defined on unit simplex (i.e., all variables are nonnegative and their sum equals one), because the computation for the moments of the Dirichlet distribution is simple. However, the Dirichlet distribution can be used only when all variables are negatively correlated. When some of the variables are significantly positively correlated, the Dirichlet distribution will be an inappropriate prior for Bayesian analysis. The multivariate distributions generated from stochastic bifurcation processes governed by independent beta laws, which will be called independent beta bifurcation distributions, allow the correlation sign to vary within a row. Hence, independent beta bifurcation distributions are more appropriate for analyzing compositional data. In this thesis, when the quantity of interest is the sum of some parameters in a vector having an independent beta bifurcation distribution, the necessary and sufficient conditions for perfect aggregation are established. The methods by considering the means together with either variances or covariances of variables to construct an independent beta bifurcation distribution are also presented.

中文
洪明君，1999 年，Dirichlet 分配,非短視行為,與演化性賽局，國立政治大學經濟學研究所碩士論文。

郭家菱，2003 年，帶有分型誤差之染病同胞對資料之貝氏調整連鎖分析方法，國立臺灣大學流行病學研究所碩士論文。

西文
Aitchison, J. (1985), A general class of distributions on the simplex, Journal of the Royal Statistical Society Series B, 136-146.

Aitchison, J. (1986), The Statistical Analysis of Compositional Data, John Wiley, New York.

Andre, H. and Philippe, F. (2002), Uncertainties in facies proportion estimation I. theoretical framework: the Dirichlet distribution, Mathematical Geology, 34(6), 679-702.

Azaiez, M. N. (1993), Perfect aggregation in reliability models with Bayesian updating, Dissertation, University of Wisconsin-Madison.

Azaiez, M. N. and Bier, V. M. (1995), Perfect aggregation in general reliability models with Bayesian updating, Applied Mathematics and Computation, 73, 281-302.

Azaiez, M. N. and Bier, V. M. (1996), Aggregation error in Bayesian analysis of reliability systems, Management Science, 42(4), 516-528.

Blackwell, D. and Girshick, M. A. (1954), Theory of Games and Statistical Decisions, John Wiley, New York.

Bouguila, N., Ziou, D., and Vaillancourt, J. (2004), Unsupervised learning of a finite mixture model based on the Dirichlet distribution and its application, IEEE Transactions on Image Processing, 13(11), 1533-1543.

Castillo, E., Hadi, A. S., and Solares, C. (1997), Learning and updating of uncertainty in Dirichlet models, Machine Learning, 26, 43-63.

Connor, R. J. and Mosimann, J. E. (1969), Concepts of independence for proportions with a generalization of the Dirichlet distribution, Journal of the American Statistical Association, 64, 194-206.

Dickey, J. M. and Jiang, T. J. (1998), Filtered-variate prior distributions for histogram smoothing, Journal of the American Statistical Association, 93(442), 651-662.

Ijiri, Y. (1971), Fundamental queries in aggregation theory, Journal of the American Statistical Association, 66, 766-782.

Iwasa, Y., Levin, S., and Andreasen, V. (1987), Aggregation in model ecosystem: I. perfect aggregation, Ecological Modelling, 37, 287-302.

Krzysztofowicz, R. and Reese, S. (1993) Stochastic bifurcation processes and distributions of fractions, Journal of the American Statistical Association, 88(421), 345-354.

Lochner, R. H. (1975), A generalized Dirichlet distribution in Bayesian life testing, Journal of the Royal Statistical Society Series B, 37, 103-113.

Mosleh, A. and Bier,V. M. (1992), On decomposition and aggregation error in estimation: some basic principles and examples, Risk Analysis, 22, 203-214.

Simon, H. and Ando A. (1961), Aggregation of variables in dynamic systems,
Econometrica, 29, 111-138.

Walley, P. (1996), Inferences from multinomial data: learning about a bag of marbles, Journal of the Royal Statistical Society Series B, 58(1), 3-34.

Wilks,S. S. (1962), Mathematical Statistics, John Wiley, New York.

Wong, T. T. (1998a), Generalized Dirichlet distribution in Bayesian analysis, Applied Mathematics and Computation, 97, 165-181.

Wong, T. T. (1998b), Perfect aggregation in dependent Bernoulli systems with Bayesian updating, Ph. D. Thesis, University of Wisconsin-Madison.

Wong, T. T. (2005), Perfect aggregation of Bayesian analysis on compositional data, Accepted by Statical Papers.