  It is common practice in statistical data analysis to perform data-driven
variable selection and derive statistical inference from the resulting model.
Such inference enjoys none of the guarantees that classical statistical theory
provides for tests and confidence intervals when the model has been chosen a
priori. We propose to produce valid ``post-selection inference'' by reducing
the problem to one of simultaneous inference and hence suitably widening
conventional confidence and retention intervals. Simultaneity is required for
all linear functions that arise as coefficient estimates in all submodels. By
purchasing ``simultaneity insurance'' for all possible submodels, the resulting
post-selection inference is rendered universally valid under all possible model
selection procedures. This inference is therefore generally conservative for
particular selection procedures, but it is always less conservative than full
Scheffe protection. Importantly it does not depend on the truth of the selected
submodel, and hence it produces valid inference even in wrong models. We
describe the structure of the simultaneous inference problem and give some
asymptotic results.
