Fisherian conditional inference

In a parametric statistical model, the likelihood function compares parameter values, while sampling distributions calibrate tests and confidence intervals. A sufficient statistic retains the parameter information. Conditioning on an observed ancillary statistic restricts that calibration to experiments with the same ancillary configuration, instead of averaging over configurations whose distribution is unrelated to the parameter. Since the ancillary's marginal law is parameter-free, conditional and unconditional likelihoods have the same parameter-dependent factor. With a nuisance parameter, a conditional law or pivotal quantity must remove it before inference for the parameter of interest can use a nuisance-free reference law.

When an experiment's configuration is observed, inference conditions on the experiment actually performed rather than averaging over its possible configurations. Conditioning on an ancillary statistic is a central application; freedom from a nuisance parameter must still be checked in the resulting conditional distribution.