  In small area estimation, it is sometimes necessary to use model-based
methods to produce estimates in areas with little or no data. In official
statistics, we often require that some aggregate of small area estimates agree
with a national estimate for internal consistency purposes. Enforcing this
agreement is referred to as benchmarking, and while methods currently exist to
perform benchmarking, few are ideal for applications with non-normal outcomes
and benchmarks with uncertainty. Fully Bayesian benchmarking is a theoretically
appealing approach insofar as we can obtain posterior distributions conditional
on a benchmarking constraint. However, existing implementations may be
computationally prohibitive. In this paper, we critically review benchmarking
methods in the context of small area estimation in low- and middle-income
countries with binary outcomes and uncertain benchmarks, and propose a novel
approach in which an unbenchmarked method that produces area-level samples can
be combined with a rejection sampler or Metropolis-Hastings algorithm to
produce benchmarked posterior distributions in a computationally efficient way.
To illustrate the flexibility and efficiency of our approach, we provide
comparisons to an existing benchmarking approach in a simulation, and
applications to HIV prevalence and under-5 mortality estimation. Code
implementing our methodology is available in the R package stbench.
