  Several new methods have been proposed for performing valid inference after
model selection. An older method is sampling splitting: use part of the data
for model selection and part for inference. In this paper we revisit sample
splitting combined with the bootstrap (or the Normal approximation). We show
that this leads to a simple, assumption-free approach to inference and we
establish results on the accuracy of the method. In fact, we find new bounds on
the accuracy of the bootstrap and the Normal approximation for general
nonlinear parameters with increasing dimension which we then use to assess the
accuracy of regression inference. We show that an alternative, called the image
bootstrap, has higher coverage accuracy at the cost of more computation. We
define new parameters that measure variable importance and that can be inferred
with greater accuracy than the usual regression coefficients. There is a
inference-prediction tradeoff: splitting increases the accuracy and robustness
of inference but can decrease the accuracy of the predictions.
