Edit: To give one example of an approach that I think is promising: We start by describing the observer and environment through a density matrix (a probability distribution over possible wave functions) and introduce an interaction with a quantum system (e.g. a spin). Given a reasonable interaction, you can show that the entanglement in the combined state (observer, environment and spin) leads to the system approaching a state that is a probability distribution of entangled states where each probability corresponds to the Born rule. Interestingly in this case the probabilities emerge from our lack of knowledge about the microstate of the observer/environment, so it’s actually thermodynamic uncertainty.