Generating versatile and appropriate synthetic speech requires control over the output expression separate from the spoken text. Important non-textual speech variation is seldom annotated, in which case output control must be learned in an unsupervised fashion. In this paper, we perform an in-depth study of methods for unsupervised learning of control in statistical speech synthesis. For example, we show that popular unsupervised training heuristics can be interpreted as variational inference in certain autoencoder models. We additionally connect these models to VQ-VAEs, another, recently-proposed class of deep variational autoencoders, which we show can be derived from a very similar mathematical argument. The implications of these new probabilistic interpretations are discussed. We illustrate the utility of the various approaches with an application to acoustic modelling for emotional speech synthesis, where the unsupervised methods for learning expression control (without access to emotional labels) are found to give results that in many aspects match or surpass the previous best supervised approach.