修正條件分配勝率矩陣時最佳參考點之選取方法

Abstract

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\n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\n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\n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\n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\n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\n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\n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\njoint 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\nconditional 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\ndiscover 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\nthen 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.