  The U.S. Census Bureau's 2020 Disclosure Avoidance System (DAS) bases its
output on noisy measurements, which are population tabulations added to
realizations of mean-zero random variables. These noisy measurements are
observed in a set of hierarchical geographic units, e.g., the U.S. as a whole,
states, counties, census tracts, and census blocks. The noisy measurements from
the 2020 Redistricting Data File and Demographic and Housing Characteristics
File statistical data products are now public. The purpose of this paper is to
describe a method to leverage the hierarchical structure within these noisy
measurements to compute confidence intervals for arbitrary tabulations and in
arbitrary geographic entities composed of census blocks. This method is based
on computing a weighted least squares estimator (WLS) and its variance matrix.
Due to the high dimension of this estimator, this operation is not feasible
using the standard approach, since this would require evaluating products with
the inverse of a dense matrix with several billion (or even several trillion)
rows and columns. In contrast, the approach we describe in this paper computes
the required estimate and its variance with a time complexity and memory
requirement that scales linearly in the number of census blocks.
