A General Space of Belief Updates for Model Misspecification in Bayesian Networks

Abstract: In an ideal setting for Bayesian agents, a perfect description of the rules of the environment (i.e., the objective observation model) is available, allowing them to reason through the Bayesian posterior to update their beliefs in an optimal way. But such an ideal setting hardly ever exists in the natural world, so agents have to make do with reasoning about how they should update their beliefs simultaneously. This introduces a number of related challenges for a number of research areas: (1) For Bayesian statistics, this deviation of the subjective model from the true data-generating mechanism is termed model misspecification in the literature. (2) For neuroscience, it introduces the necessity to model how the agents' belief updates (how they use evidence to update their belief) and how their belief changes over time. The current paper addresses these two challenges by (a) providing a general class of posteriors/belief updates called cut-posteriors of Bayesian networks that have a much greater expressivity, and (b) parameterizing the space of possible posteriors to make meta-learning (i.e., choosing the belief update from this space in a principled manner) possible. For (a), it is noteworthy that any cut-posterior has local computation only, making computation tractable for human or artificial agents. For (b), a Markov Chain Monte Carlo algorithm to perform such meta-learning will be sketched here, though it is only an illustration and but no means the only possible meta-learning procedure possible for the space of cut-posteriors. Operationally, this work gives a general algorithm to take in an arbitrary Bayesian network and output all possible cut-posteriors in the space.