Machine learning methods often fail when deployed in the real world. Worse still, they fail in high-stakes situations and across socially sensitive lines. These issues have a chilling effect on the adoption of machine learning methods in settings such as medical diagnosis, where they are arguably best-placed to provide benefits if safely deployed. In this primer, we introduce the causal and statistical structures which induce failure in machine learning methods for image analysis. We highlight two previously overlooked problems, which we call the \textit{no fair lunch} problem and the \textit{subgroup separability} problem. We elucidate why today's fair representation learning methods fail to adequately solve them and propose potential paths forward for the field.