The Roles of Conditioning in Inference

This paper reviews the use of sufficient and ancillary statistics in constructing conditional distributions for inference about a parameter. Special emphasis is given to recent developments in accurate approximation of densities, distribution functions and likelihood functions, and to the role of conditioning in these approximations. Exact conditional or marginal inference is available for essentially two classes of models, exponential family models and transformation family models. The approximations are very useful for practical implementation of these exact results. The form of the approximations suggests methods for inference in more general families.