In the Principles and Parameters framework, the structural features of languages depend on parameters that may be toggled on or off, with a single parameter often dictating the status of multiple features. The implied covariance between features inspires our probabilisation of this line of linguistic inquiry—we develop a generative model of language based on exponential-family matrix factorisation. By modelling all languages and features within the same architecture, we show how structural similarities between languages can be exploited to predict typological features with near-perfect accuracy, besting several baselines on the task of predicting held-out features. Furthermore, we show that language representations pre-trained on monolingual text allow for generalisation to unobserved languages. This finding has clear practical and also theoretical implications: the results confirm what linguists have hypothesised, i.e. that there are significant correlations between typological features and languages.