  The success of neural networks on a diverse set of NLP tasks has led
researchers to question how much these networks actually ``know'' about natural
language. Probes are a natural way of assessing this. When probing, a
researcher chooses a linguistic task and trains a supervised model to predict
annotations in that linguistic task from the network's learned representations.
If the probe does well, the researcher may conclude that the representations
encode knowledge related to the task. A commonly held belief is that using
simpler models as probes is better; the logic is that simpler models will
identify linguistic structure, but not learn the task itself. We propose an
information-theoretic operationalization of probing as estimating mutual
information that contradicts this received wisdom: one should always select the
highest performing probe one can, even if it is more complex, since it will
result in a tighter estimate, and thus reveal more of the linguistic
information inherent in the representation. The experimental portion of our
paper focuses on empirically estimating the mutual information between a
linguistic property and BERT, comparing these estimates to several baselines.
We evaluate on a set of ten typologically diverse languages often
underrepresented in NLP research---plus English---totalling eleven languages.
