Elimination of Nuisance Parameters

The problem begins with an unknown state of nature represented by the parameter of interest θ . We have some information about θ to begin with — e.g., we know that θ is a member of some well-defined parameter space θ- but we are seeking more. Toward this end, a statistical experiment & is planned and performed and this generates the sample observation x. Further information about θ is then obtained by a careful analysis of the data ( &, x) in the light of all our prior information about θ and in the context of the particular inference problem related to θ . For going through the rituals of the traditional sample-space analysis of data, we must begin with the invocation of a trinity of abstractions ( X, A, P ), where X is the sample space, A is a σ-algebra of events (subsets of X ), and P is a family of probability measures on A . If the model (X, A, P) is such that we can represent the family P as {P_θ: θ εθ}, where the correspondence θ → P_θ is one-one and (preferably) smooth, then we go about analyzing the data according to our own light and are thankful for not having to contend with any nuisance parameters.