  Zipf (1935) posited that wordforms are optimized to minimize utterances'
communicative costs. Under the assumption that cost is given by an utterance's
length, he supported this claim by showing that words' lengths are inversely
correlated with their frequencies. Communicative cost, however, can be
operationalized in different ways. Piantadosi et al. (2011) claim that cost
should be measured as the distance between an utterance's information rate and
channel capacity, which we dub the channel capacity hypothesis (CCH) here.
Following this logic, they then proposed that a word's length should be
proportional to the expected value of its surprisal (negative log-probability
in context). In this work, we show that Piantadosi et al.'s derivation does not
minimize CCH's cost, but rather a lower bound, which we term CCH-lower. We
propose a novel derivation, suggesting an improved way to minimize CCH's cost.
Under this method, we find that a language's word lengths should instead be
proportional to the surprisal's expectation plus its variance-to-mean ratio.
Experimentally, we compare these three communicative cost functions: Zipf's,
CCH-lower , and CCH. Across 13 languages and several experimental settings, we
find that length is better predicted by frequency than either of the other
hypotheses. In fact, when surprisal's expectation, or expectation plus
variance-to-mean ratio, is estimated using better language models, it leads to
worse word length predictions. We take these results as evidence that Zipf's
longstanding hypothesis holds.
