Since convolutional neural networks (CNNs) achieved top performance on the ImageNet task in 2012, deep learning has become the preferred approach to addressing computer vision, natural language processing, speech recognition and bio-informatics tasks. However, despite impressive performance, neural networks tend to make over-confident predictions. Thus, it is necessary to investigate robust, interpretable and tractable estimates of uncertainty in a model's predictions in order to construct

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... Machine Learning systems. This is crucial to applications where the cost of an error is high, such as in autonomous vehicle control, high-stakes automatic proficiency assessment and in the medical, financial and legal fields. In the first part of this thesis uncertainty estimation via ensemble and single-model approaches is discussed in detail and a new class of models for uncertainty estimation, called $\textit{Prior Networks}$, is proposed. Prior Networks are able to $\textit{emulate}$ an ensemble of models using a single deterministic neural network, which allows sources of uncertainty to be determined within the same probabilistic framework as in ensemble-based approaches, but with the computational simplicity and ease of training of single-model approaches. Thus, Prior Networks combine the advantages of ensemble and single-model approaches to estimating uncertainty. In this thesis Prior Networks are evaluated on a range classification datasets, where they are shown to outperform baseline approaches, such as Monte-Carlo dropout, on the task of detecting out-of-distribution inputs. In the second part of this thesis deep learning and uncertainty estimation approaches are applied to the area of automatic assessment of non-native spoken language proficiency. Specifically deep-learning based graders and spoken response relevance assessment systems are constructed using data from the BULATS and LinguaSkill exams, provided by Cambridge English Language Assessment. Baseline approaches for uncertainty estimation discussed and [...]