二維條件分配相容性問題之新解法

Abstract

給定二元隨機向量(X,Y)之聯合機率分配，可容易得到其條件機率分配X|Y與Y|X；反之，給定條件機率分配X|Y與Y|X，是否能獲得對應的聯合機率分配呢？條件分配相容性研究的主要內容包括：(一)如何判斷給定的條件分配是否相容？(二)若相容，則如何找到聯合分配？(三)若不相容，則該如何找到最近似的聯合分配？\n\n根據比值矩陣法的理論，檢驗比值矩陣是否為秩1矩陣或者有擴張秩1矩陣，便可得知給定的條件分配是否相容。當比值矩陣的元素皆為正值時，本文運用線性代數中之奇異值分解定理，先發展出奇異值分解法來處理條件分配相容性問題；當比值矩陣的元素非皆為正值時，接續發展出最近似秩1矩陣法來解決相容性問題，而最近似秩1矩陣法可視為奇異值分解法的延伸。在發展最近似秩1矩陣法時，我們利用到類Frobenius範數的概念，並提出了三種求解過程(無限制條件法、Lagrange乘數法與高維度牛頓法)以及相關的演算法。本文詳細剖析了三種求解過程之數學流程，並輔以實際例子予以說明。\n\n當條件機率分配不相容時，我們通常可獲得兩組近似聯合分配。如何將它們做適當的組合，也是值得探討的問題。最後，針對等加權之組合方式、權重與總誤差成反比之組合方式以及特徵向量法之組合方式進行比較分析。

Given a bivariate joint distribution of random vector (X,Y), we can easily derive the conditional probability distributions of X|Y and Y|X. Conversely, given conditional probability distributions of X|Y and Y|X, can we find the corresponding joint distribution? The compatibility issues of conditional distribution include: (a) how to determine whether they are compatible; (b) how to find the joint distribution if they are compatible; (c) how to find the most nearly joint distribution if they are incompatible.\n\nUsing the theory of ratio matrix approach, we can determine the given conditional probability distributions are compatible or not by checking whether their corresponding ratio matrix or the extension matrix of this ratio matrix is rank one or not. When elements of the ratio matrix are all positive, this thesis uses the singular value decomposition theorem of linear algebra to develop the singular value decomposition approach to deal with the compatibility issues. When elements of the ratio matrix are not all positive, we provide the most nearly rank one matrix approach to solve the compatibility issues. This most nearly rank one matrix approach can be considered as the extension of singular value decomposition approach. To develop the most nearly rank one matrix approach, we use the concept of semi-Frobenius norm to provide three solving methods (unconstrained method, Lagrange multiplier method, and multivariate Newton`s method) with related algorithms. This thesis gives the mathematical procedure on these three solving methods in detail and uses examples to explain the compatibility issues.\n\nWhen the conditional distributions are incompatible, we usually have two nearly joint distributions. It would be worth of discussing the combination of these two nearly joint distributions. Hence, this thesis compares and analyzes the compatibility issues with three different weights, which are equal, inverse proportional to the total errors, and relating to eigenvectors.