@MoreAnonymous Sure, there's nothing wrong with it - I don't think the answer is going to be anything insightful - at first glance it looks like an in principle standard application of Bayes rule.

There are a couple of differences in that I'd argue you should talk about $t+\delta t$, as it's continuous and that you'd need to take any measurements into account, which makes things much more interesting

and you should really talk about either 'we have a state that's continuously evolving and we're not measuring it, so this is what the probability would be, if we measured it' or 'we made a series of measurements and have the probabilities for each measurement etc.' because these are slightly different things, really

Slightly different in that the first is continuous, the second is discrete

@Mithrandir24601 Good advice and took it :D