> I'm not able to understand why any of that is bothersome.

Ok. It's bothersome to me because Bayes gets trotted out in the context of Kalman filters as though it is shedding light on the topic or providing rigor to the approach, but I haven't found anyone who can really describe what the elements of that expression are without hand waiving.

You've switched from capital 'P' to lowercase 'p' in your message (which I take to mean Probability and pdf respectively). If that was intentional, it opens a new batch of questions for what the notation means.

Anyways, maybe I'm just a slow learner, too pedantic, or something else. If you're interested, read my other reply where I go into it more.