  To measure how well pretrained representations encode some linguistic
property, it is common to use accuracy of a probe, i.e. a classifier trained to
predict the property from the representations. Despite widespread adoption of
probes, differences in their accuracy fail to adequately reflect differences in
representations. For example, they do not substantially favour pretrained
representations over randomly initialized ones. Analogously, their accuracy can
be similar when probing for genuine linguistic labels and probing for random
synthetic tasks. To see reasonable differences in accuracy with respect to
these random baselines, previous work had to constrain either the amount of
probe training data or its model size. Instead, we propose an alternative to
the standard probes, information-theoretic probing with minimum description
length (MDL). With MDL probing, training a probe to predict labels is recast as
teaching it to effectively transmit the data. Therefore, the measure of
interest changes from probe accuracy to the description length of labels given
representations. In addition to probe quality, the description length evaluates
"the amount of effort" needed to achieve the quality. This amount of effort
characterizes either (i) size of a probing model, or (ii) the amount of data
needed to achieve the high quality. We consider two methods for estimating MDL
which can be easily implemented on top of the standard probing pipelines:
variational coding and online coding. We show that these methods agree in
results and are more informative and stable than the standard probes.
