We study when logged contextual-bandit data can certify an upper tail quantile for a policy selected from the same sample.
We construct a PAC-Bayes Bernstein lower-confidence envelope for the target cumulative distribution function (CDF), simultaneous over a predeclared threshold grid and all data-dependent posteriors.
The resulting procedure, TailCert, uses the envelope's first crossing of level p to return an atom-safe upper confidence bound on the selected posterior's pooled p-quantile (or inconclusive).
It can also be used to select a policy under a certified tail constraint.
The concentration ingredients are standard; the main theoretical contribution is a matched finite-class certification boundary comparing full logs with their joint weighted CDF indicators at a fixed threshold.
On an explicit rare-action class, the required sample size grows linearly with the overlap factor W, logarithmically with policy-search and confidence complexity, and quadratically with the inverse CDF safety margin; the upper and lower bounds agree up to constants in the stated regime.
On this class, the CDF reduction hides which actions exceeded the budget; retaining action counts restores the full-log rate.
Simulations verify the predicted search-complexity shift, winner's-curse failure of naive reuse, and the power cost of splitting.