  Donald Trump was lagging behind in nearly all opinion polls leading up to the
2016 US presidential election, but he surprisingly won the election. This
raises the following important questions: 1) why most opinion polls were not
accurate in 2016? and 2) how to improve the accuracies of opinion polls? In
this paper, we study the inaccuracies of opinion polls in the 2016 election
through the lens of information theory. We first propose a general framework of
parameter estimation, called clean sensing (polling), which performs optimal
parameter estimation with sensing cost constraints, from heterogeneous and
potentially distorted data sources. We then cast the opinion polling as a
problem of parameter estimation from potentially distorted heterogeneous data
sources, and derive the optimal polling strategy using heterogenous and
possibly distorted data under cost constraints. Our results show that a larger
number of data samples do not necessarily lead to better polling accuracy,
which give a possible explanation of the inaccuracies of opinion polls in 2016.
The optimal sensing strategy should instead optimally allocate sensing
resources over heterogenous data sources according to several factors including
data quality, and, moreover, for a particular data source, it should strike an
optimal balance between the quality of data samples, and the quantity of data
samples.
  As a byproduct of this research, in a general setting, we derive a group of
new lower bounds on the mean-squared errors of general unbiased and biased
parameter estimators. These new lower bounds can be tighter than the classical
Cram\'{e}r-Rao bound (CRB) and Chapman-Robbins bound. Our derivations are via
studying the Lagrange dual problems of certain convex programs. The classical
Cram\'{e}r-Rao bound and Chapman-Robbins bound follow naturally from our
results for special cases of these convex programs.
