  Pimentel et al. (2020) recently analysed probing from an
information-theoretic perspective. They argue that probing should be seen as
approximating a mutual information. This led to the rather unintuitive
conclusion that representations encode exactly the same information about a
target task as the original sentences. The mutual information, however, assumes
the true probability distribution of a pair of random variables is known,
leading to unintuitive results in settings where it is not. This paper proposes
a new framework to measure what we term Bayesian mutual information, which
analyses information from the perspective of Bayesian agents -- allowing for
more intuitive findings in scenarios with finite data. For instance, under
Bayesian MI we have that data can add information, processing can help, and
information can hurt, which makes it more intuitive for machine learning
applications. Finally, we apply our framework to probing where we believe
Bayesian mutual information naturally operationalises ease of extraction by
explicitly limiting the available background knowledge to solve a task.
