A characteristic of most real world problems is that collecting unlabelled examples is easier and cheaper than collecting labelled ones. As a result, learning from partially labelled data is a crucial and demanding area of machine learning, and extending techniques from fully to partially supervised scenarios is a challenging problem. Our work focuses on two types of partially labelled data that can occur in binary problems: semi-supervised data, where the labelled set contains both positive and negative examples, and positive-unlabelled data, a more restricted version of partial supervision where the labelled set consists of only positive examples. In both settings, it is very important to explore a large number of features in order to derive useful and interpretable information about our classification task, and select a subset of features that contains most of the useful information.In this thesis, we address three fundamental and tightly coupled questions concerning feature selection in partially labelled data; all three relate to the highly controversial issue of when does additional unlabelled data improve performance in partially labelled learning environments and when does not. The first question is what are the properties of statistical hypothesis testing in such data? Second, given the widespread criticism of significance testing, what can we do in terms of effect size estimation, that is, quantification of how strong the dependency between feature X and the partially observed label Y? Finally, in the context of feature selection, how well can features be ranked by estimated measures, when the population values are unknown? The answers to these questions provide a comprehensive picture of feature selection in partially labelled data. Interesting applications include for estimation of mutual information quantities, structure learning in Bayesian networks, and investigation of how human-provided prior knowledge can overcome the restrictions of partial labelling.One direct contribution of our work is to enable valid statistical hypothesis testing and estimation in positive-unlabelled data. Focusing on a generalised likelihood ratio test and on estimating mutual information, we provide five key contributions. (1) We prove that assuming all unlabelled examples are negative cases is sufficient for independence testing, but not for power analysis activities. (2) We suggest a new methodology that compensates this and enables power analysis, allowing sample size determination for observing an effect with a desired power by incorporating user's prior knowledge over the prevalence of positive examples. (3) We show a new capability, supervision determination, which can determine a-priori the number of labelled examples the user must collect before being able to observe a desired statistical effect. (4) We derive an estimator of the mutual information in positive-unlabelled data, and its asymptotic distribution. (5) Finally, we show how to rank features with and without prior knowledge. Also we derive extensions of these results to semi-supervised data.In another extension, we investigate how we can use our results for Markov blanket discovery in partially labelled data. While there are many different algorithms for deriving the Markov blanket of fully supervised nodes, the partially labelled problem is far more challenging, and there is a lack of principled approaches in the literature. Our work constitutes a generalization of the conditional tests of independence for partially labelled binary target variables, which can handle the two main partially labelled scenarios: positive-unlabelled and semi-supervised. The result is a significantly deeper understanding of how to control false negative errors in Markov Blanket discovery procedures and how unlabelled data can help.Finally, we present how our results can be used for information theoretic feature selection in partially labelled data. Our work extends naturally feature selection criteria suggested for fully-supervised data, to partially labelled scenarios. These criteria can capture both the relevancy and redundancy of the features and can be used for semi-supervised and positive-unlabelled data.