  Investigators often use the data to generate interesting hypotheses and then
perform inference for the generated hypotheses. P-values and confidence
intervals must account for this explorative data analysis. A fruitful method
for doing so is to condition any inferences on the components of the data used
to generate the hypotheses, thus preventing information in those components
from being used again. Some currently popular methods "over-condition", leading
to wide intervals. We show how to perform the minimal conditioning in a
computationally tractable way. In high dimensions, even this minimal
conditioning can lead to intervals that are too wide to be useful, suggesting
that up to now the cost of hypothesis generation has been underestimated. We
show how to generate hypotheses in a strategic manner that sharply reduces the
cost of data exploration and results in useful confidence intervals. Our
discussion focuses on the problem of post-selection inference after fitting a
lasso regression model, but we also outline its extension to a much more
general setting.
