ABSTRACT
Virtually every practical problem in statistics involves more than one unknown or unobservable quantity. It is in dealing with such problems that the simple conceptual framework of the Bayesian approach reveals its principal advantages over other methods of inference. Although a problem can include several parameters of interest, conclusions will often be drawn about one, or only a few, parameters at a time. In this case, the ultimate aim of a Bayesian analysis is to obtain the marginal posterior distribution of the particular parameters of interest. In principle, the route to achieving this aim is clear: we first require the joint posterior distribution of all unknowns, and then we integrate this distribution over the unknowns that are not of immediate interest to obtain the desired marginal distribution. Or equivalently, using simulation, we draw samples from the joint posterior distribution and then look at the parameters of interest and ignore the values of the other unknowns. In many problems there is no interest in making inferences about many of the unknown parameters, although they are required in order to construct a realistic model. Parameters of this kind are often called nuisance parameters. A classic example is the scale of the random errors in a measurement problem.
