论文标题
最佳的“和”
The Optimal 'AND'
论文作者
论文摘要
联合分销$ p（x，y）$无法从其边际$ p（x）$和$ p（y）$的$中确定；一个人还需要其中一个条件$ p（x | y）$或$ p（y | x）$。但是，只有边缘人只有最好的猜测？在这里，我们以肯定的形式回答这个问题，以封闭形式获得边缘的功能，该功能在未知的“ True”关节概率和功能值之间具有最低的预期Kullbach-Liebler（KL）差异。考虑到边际，在可能的关节概率值上，杰弗里斯的非信息性先验是对期望的。此分布也可用于获得任何其他“聚合操作员”的预期信息损失，因为对于任何给定的一对边缘输入值，这种估计器通常以模糊逻辑为单位。这使得在我们假设的最低知识条件下的预期损失中，可以根据其预期的损失进行比较。我们继续开发一种方法，以评估任何聚合运算符的预期准确性，而没有其输入知识。这需要在所有可能的输入对上平均预期损失，并通过适当的分布加权。我们通过将Jeffreys在可能的关节分布（在两个布尔变量上的关节分布空间的3个功能独立坐标上）进行边缘化来获得此分布，这是一个边缘分布对的关节分布，这是一个边缘分布的一个二维空间，每个边际分布都有一个参数。我们报告了一些常用的运营商以及最佳操作员的结果输入平均预期损失。最后，我们讨论了将我们的方法发展为一种有原则的风险管理方法的潜力，以替换针对概率图形模型做出的通常是任意的条件独立假设。
The joint distribution $P(X,Y)$ cannot be determined from its marginals $P(X)$ and $P(Y)$ alone; one also needs one of the conditionals $P(X|Y)$ or $P(Y|X)$. But is there a best guess, given only the marginals? Here we answer this question in the affirmative, obtaining in closed form the function of the marginals that has the lowest expected Kullbach-Liebler (KL) divergence between the unknown "true" joint probability and the function value. The expectation is taken with respect to Jeffreys' non-informative prior over the possible joint probability values, given the marginals. This distribution can also be used to obtain the expected information loss for any other "aggregation operator", as such estimators are often called in fuzzy logic, for any given pair of marginal input values. This enables such such operators, including ours, to be compared according to their expected loss under the minimal knowledge conditions we assume. We go on to develop a method for evaluating the expected accuracy of any aggregation operator in the absence of knowledge of its inputs. This requires averaging the expected loss over all possible input pairs, weighted by an appropriate distribution. We obtain this distribution by marginalizing Jeffreys' prior over the possible joint distributions (over the 3 functionally independent coordinates of the space of joint distributions over two Boolean variables) onto a joint distribution over the pair of marginal distributions, a 2-dimensional space with one parameter for each marginal. We report the resulting input-averaged expected losses for a few commonly used operators, as well as the optimal operator. Finally, we discuss the potential to develop our methodology into a principled risk management approach to replace the often rather arbitrary conditional-independence assumptions made for probabilistic graphical models.
