There is an ongoing debate in the NLP community whether modern language models contain linguistic knowledge, recovered through so-called textit{probes}. In this paper we study whether linguistic knowledge is a necessary condition for good performance of modern language models, which we call the textit{rediscovery hypothesis}. In the first place we show that language models that are significantly compressed but perform well on their pretraining objectives retain good scores when probed for linguistic structures. This result supports the rediscovery hypothesis and leads to the second contribution of our paper: an information-theoretic framework that relates language modeling objective with linguistic information. This framework also provides a metric to measure the impact of linguistic information on the word prediction task. We reinforce our analytical results with various experiments, both on synthetic and on real tasks.
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