  Post-selection inference has recently been proposed as a way of quantifying
uncertainty about detected changepoints. The idea is to run a changepoint
detection algorithm, and then re-use the same data to perform a test for a
change near each of the detected changes. By defining the p-value for the test
appropriately, so that it is conditional on the information used to choose the
test, this approach will produce valid p-values. We show how to improve the
power of these procedures by conditioning on less information. This gives rise
to an ideal selective p-value that is intractable but can be approximated by
Monte Carlo. We show that for any Monte Carlo sample size, this procedure
produces valid p-values, and empirically that noticeable increase in power is
possible with only very modest Monte Carlo sample sizes. Our procedure is easy
to implement given existing post-selection inference methods, as we just need
to generate perturbations of the data set and re-apply the post-selection
method to each of these. On genomic data consisting of human GC content, our
procedure increases the number of significant changepoints that are detected
from e.g. 17 to 27, when compared to existing methods.
