Ancillary Statistics, Pivotal Quantities and Confidence Statements

The most commonly used expression in Statistics is information; yet, we have no agreement on the definition or usage of this concept. However, in the particular situation where the problem is to predict a future value of a random variable X with a known probability distribution p(.), we all seem to agree that the information on the yet unobserved future value of X may be characterized by the function p(.) itself. And if we have another variable Y such that the conditional distribution p(. |Y) of X, given Y, is also known then, having observed Y, we can claim that the information on X has shifted from p(.) to p(. | Y). [To avoid a multiplicity of notations, we do not distinguish between a random variable X, an observed value of X and a typical point in the sample space of X.] If p(.|Y)is the same for all values of Y, then X is stochastically independent of Y. In this case Y is said to have no information on X. And we know how to prove then that X has no information on Y.