This reminds me of this passage from Richard Jeffrey’s The Logic of Decision:
Socrates argues (Symposium, 200) that Love is not a god because to desire something is to be in want of it: you cannot desire what you already have. The point is also made in contemporary dictionaries. Thus, The Concise Oxford Dictionary defines a desire as an unsatisfied appetite, longing, wish, or craving. This account of the matter seems roughly right, although it needs some refinement. Thus, to vary the example of the Symposium, since people do not always know when they are loved, it is entirely possible to desire someone’s love when you already have it. Therefore, it seems better to say that you cannot desire what you think you have. Taking propositions as the objects of desire, the doctrine becomes: one who believes that a proposition is true cannot desire that it be true.
Which does suggest that both believing and wanting is related to reducing prediction error, because wanting something is wanting it to be true, and wanting something to be true is wanting to increase your degree of belief that it is true. So they are indeed related but they don’t seem hard to distinguish.
Can you make the relevance of this research more tangible by giving an informal example of Alice and Bob “observing” two different correlated variables, as well as a shared “latent variable”? I have difficulty imagining what those variables would model in practice.