Suppose we have two random variables $p \in [0,1]$ and $q \in [0,1]$ with correlation $\rho$. My question: knowing $\rho$, how do I update my probability distribution over values of one of the variables, when I learn the value of the other?
As an example, suppose that $\rho = 0.5$, and I learn that $p=1$. What can I say about the expected value $q$? If I'm maximizing accuracy, what value should I predict? Plausibly, if the expected value of $q$ was $0.5$ before (perhaps by the principle of insufficient reason) then I should adjust this upwards when I learn of the correlation and value of $p$. But in what way, and how much exactly?
I think I'm confused about how correlation relates to probabilities and Bayesian updating. Any help would be much appreciated!