Representation learning is a process that an agent uses to alter which pieces of information it retains from its input stream, and which pieces it discards, implementing an intelligent compression scheme. There are strong theoretical underpinnings for using Bayesian inference for compression problems, and also for learning to make good decisions based on a pre-existing representation, but there is currently no theory describing how representations should change over time in a task-based setting, which would be of major interest to both cognitive science and machine learning. We want to develop such a theory by formalising the interplay between the inductive biases of the agent, the constraints on available resources and the task objective in a learning algorithm. The student will participate in the algorithmic design of agents and tasks testing the theory, the implementation of the tree-search based and gradient-based algorithmic components, and the evaluation of the results.