Eliminating nuisance parameters from a model is universally recognized as a major problem of statistics. A surprisingly large number of elimination methods have been proposed by various writers on the topic. In this article we propose to critically review two such elimination methods. We shall be concerned with some particular cases of the marginalizing and the conditioning methods. The origin of these methods may be traced to the work of Sir Ronald A. Fisher. The contents of the marginalization and the conditionality arguments are then reexamined from the Bayesian point of view. This article should be regarded as a sequel to the author's three-part essay (Basu 1975) on statistical information and likelihood.