  Computational pseudorandomness studies the extent to which a random variable
$\bf{Z}$ looks like the uniform distribution according to a class of tests
$\cal{F}$. Computational entropy generalizes computational pseudorandomness by
studying the extent which a random variable looks like a \emph{high entropy}
distribution. There are different formal definitions of computational entropy
with different advantages for different applications. Because of this, it is of
interest to understand when these definitions are equivalent.
  We consider three notions of computational entropy which are known to be
equivalent when the test class $\cal{F}$ is closed under taking majorities.
This equivalence constitutes (essentially) the so-called \emph{dense model
theorem} of Green and Tao (and later made explicit by Tao-Zeigler, Reingold et
al., and Gowers). The dense model theorem plays a key role in Green and Tao's
proof that the primes contain arbitrarily long arithmetic progressions and has
since been connected to a surprisingly wide range of topics in mathematics and
computer science, including cryptography, computational complexity,
combinatorics and machine learning. We show that, in different situations where
$\cal{F}$ is \emph{not} closed under majority, this equivalence fails. This in
turn provides examples where the dense model theorem is \emph{false}.
