  We extend methods of Ding and Smart from their breakthrough paper in 2020
which showed Anderson localization for certain random Schr\"odinger operators
on $\ell^2(\mathbb{Z}^2)$ via a quantitative unique continuation principle and
Wegner estimate. We replace the requirement of identical distribution with the
requirement of a uniform bound on the essential range of potential and a
uniform positive lower bound on the variance of the variables giving the
potential. Under those assumptions, we recover the unique continuation and
Wegner lemma results, using Bernoulli decompositions and modifications of the
arguments therein. This leads to a localization result at the bottom of the
spectrum.
