Anchored Packed Trees (APTs) are a novel approach to distributional semantics that takes distributional composition to be a process of lexeme contextualisation. A lexeme's meaning, characterised as knowledge concerning co-occurrences involving that lexeme, is represented with a higher-order dependency-typed structure (the APT) where paths associated with higher-order dependencies connect vertices associated with weighted lexeme multisets. The central innovation in the compositional theory is that the APT's type structure enables the precise alignment of the semantic representation of each of the lexemes being composed. Like other count-based distributional spaces, however, Anchored Packed Trees are prone to considerable data sparsity, caused by not observing all plausible co-occurrences in the given data. This problem is amplified for models like APTs, that take the grammatical type of a co-occurrence into account. This results in a very sparse distributional space, requiring a mechanism for inferring missing knowledge. Most methods face this challenge in ways that render the resulting word representations uninterpretable, with the consequence that distributional composition becomes difficult to model and reason about. In this thesis, I will present a practical evaluation of the Apt theory, including a large-scale hyperparameter sensitivity study and a characterisation of the distributional space that APTs give rise to. Based on the empirical analysis, the impact of the problem of data sparsity is investigated. In order to address the data sparsity challenge and retain the interpretability of the model, I explore an alternative algorithm — distributional inference — for improving elementary representations. The algorithm involves explicitly inferring unobserved co-occurrence events by leveraging the distributional neighbourhood of the semantic space. I then leverage the rich type structure in APTs and propose a generalisation of the distributional inference algorithm. I empirically show that distributional inference improves elementary word representations and is especially beneficial when combined with an intersective composition function, which is due to the complementary nature of inference and composition. Lastly, I qualitatively analyse the proposed algorithms in order to characterise the knowledge that they are able to infer, as well as their impact on the distributional APT space.