In historical linguistics, cognate detection is the task of determining whether sets of words have common etymological roots. Inspired by the comparative method used by human linguists, we develop a system for automated cognate detection that frames the task as an inference problem for a general statistical model consisting of observed data (potentially cognate pairs of words), latent variables (the cognacy status of pairs) and unknown global parameters (which sounds correspond between languages). We then give a specific instance of such a model along with an expectation-maximisation algorithm to infer its parameters. We evaluate our system on a corpus of 8140 cognate sets, finding the performance of our method to be comparable to the state of the art. We additionally carry out qualitative analysis demonstrating advantages it has over existing systems. We also suggest several ways our work could be extended within the general theoretical framework we propose.