  Constant entropy rate (conditional entropies must remain constant as the
sequence length increases) and uniform information density (conditional
probabilities must remain constant as the sequence length increases) are two
information theoretic principles that are argued to underlie a wide range of
linguistic phenomena. Here we revise the predictions of these principles to the
light of Hilberg's law on the scaling of conditional entropy in language and
related laws. We show that constant entropy rate (CER) and two interpretations
for uniform information density (UID), full UID and strong UID, are
inconsistent with these laws. Strong UID implies CER but the reverse is not
true. Full UID, a particular case of UID, leads to costly uncorrelated
sequences that are totally unrealistic. We conclude that CER and its particular
cases are incomplete hypotheses about the scaling of conditional entropies.
