  Summaries of massive data sets support approximate query processing over the
original data. A basic aggregate over a set of records is the weight of
subpopulations specified as a predicate over records' attributes. Bottom-k
sketches are a powerful summarization format of weighted items that includes
priority sampling and the classic weighted sampling without replacement. They
can be computed efficiently for many representations of the data including
distributed databases and data streams.
  We derive novel unbiased estimators and efficient confidence bounds for
subpopulation weight. Our estimators and bounds are tailored by distinguishing
between applications (such as data streams) where the total weight of the
sketched set can be computed by the summarization algorithm without a
significant use of additional resources, and applications (such as sketches of
network neighborhoods) where this is not the case.
  Our rigorous derivations are based on clever applications of the
Horvitz-Thompson estimator, and are complemented by efficient computational
methods. We demonstrate their benefit on a wide range of Pareto distributions.
