Informatics paradigms for brain and mental health research have seen significant advances in recent years. These developments can largely be attributed to the emergence of new technologies such as machine learning, deep learning, and artificial intelligence. Data-driven methods have the potential to support mental health care by providing more precise and personalised approaches to detection, diagnosis, and treatment of depression. In particular, precision psychiatry is an emerging field that utilises advanced computational techniques to achieve a more individualised approach to mental health care. This survey provides an overview of the ways in which artificial intelligence is currently being used to support precision psychiatry. Advanced algorithms are being used to support all phases of the treatment cycle. These systems have the potential to identify individuals suffering from mental health conditions, allowing them to receive the care they need and tailor treatments to individual patients who are mostly to benefit. Additionally, unsupervised learning techniques are breaking down existing discrete diagnostic categories and highlighting the vast disease heterogeneity observed within depression diagnoses. Artificial intelligence also provides the opportunity to shift towards evidence-based treatment prescription, moving away from existing methods based on group averages. However, our analysis suggests there are several limitations currently inhibiting the progress of data-driven paradigms in care. Significantly, none of the surveyed articles demonstrate empirically improved patient outcomes over existing methods. Furthermore, greater consideration needs to be given to uncertainty quantification, model validation, constructing interdisciplinary teams of researchers, improved access to diverse data and standardised definitions within the field. Empirical validation of computer algorithms via randomised control trials which demonstrate measurable improvement to patient outcomes are the next step in progressing models to clinical implementation.