I've encountered this problem but my stats is rusty and I've been having trouble formalizing it enough to tackle it.
I have something like a family tree where each node in the tree has two parents and can be one of two parents for some other node(s) (or not if it's a leaf node). Each node is either red or black and inherited its color from one of its parents, chosen randomly with equal probability of being from either parent.
Each node has a prior probability for being red (based on some educated guesses). I'm then told the color of a certain subset of nodes. I want to update my tree with posterior probabilities given the new information. That is, the nodes I'm given have their prosterior probabilities to be red set to either 1 or 0 and the unknown node probabilities are updated to incorporate the new information.
As an example, this tree has some known nodes (the ones colored red or black) and some unknown nodes (the blank ones). I want to update the probabilities of the unknown nodes given the known nodes.
This feels like an obvious problem for Bayesian analysis, and it's very similar to something like this, but the simultaneous nature of updating the posterior probabilities for all nodes in the tree is confusing me and I haven't figured out how to set up the update.