  The classical correlation inequality of Harris asserts that any two monotone
increasing families on the discrete cube are nonnegatively correlated. In 1996,
Talagrand established a lower bound on the correlation in terms of how much the
two families depend simultaneously on the same coordinates. Talagrand's method
and results inspired a number of important works in combinatorics and
probability theory.
  In this paper we present stronger correlation lower bounds that hold when the
increasing families satisfy natural regularity or symmetry conditions. In
addition, we present several new classes of examples for which Talagrand's
bound is tight.
  A central tool in the paper is a simple lemma asserting that for monotone
events noise decreases correlation. This lemma gives also a very simple
derivation of the classical FKG inequality for product measures, and leads to a
simplification of part of Talagrand's proof.
