Inversion of Bayes formula and measures of Bayesian information gain and pairwise dependence

By inverting the Bayes formula in a point-wise manner, we develop measures quantifying the information gained by the Bayesian process, in reference to the Fisher information. Simple examples are used for focused illustrations of the ideas. Numerical computation for the measures is discussed with formulae. By extending the information gain concept to the broader context of distribution theory, we arrive at a pairwise dependence measure, which can handle the case of functional dependence and becomes Pearson's phi(2) when the joint probability density function is defined.