Many systems can be described in a deterministic fashion. That is to say, given some collection of information about a system, you may reliably and consistently predict some other information about the system.

No, not that posterior, this one: $p(v|u)$. The probability of causes given outcomes. Finding such a probability, or even estimating it, lies at the crux of many learning algorithms. And one approach to finding such a posterior is via the use of generative models.