An exploration of parameter duality in statistical inference

Abstract

Well-known debates among statistical inferential paradigms emerge from conflicting views on the notion of probability. One dominant view understands probability as a representation of sampling variability, another prominent view understands probability as a measure of belief. The former generally describes model parameters as fixed values, in contrast to the latter. We propose there are actually two versions of a parameter within both paradigms: a fixed, unknown value which generated the data and a random version to describe the uncertainty in estimating the unknown value. An inferential approach based on confidence distributions deciphers seemingly conflicting perspectives on parameters and probabilities.