When I complain about the Bayes theorem way of describing or thinking about Kalman filters, I really mean that I don't understand what almost anything in that expression means. The notation is intuitive if read as English, but opaque to what operations are being performed on what mathematical objects. Capital P means probability of an event or predicate, and A is some abstraction of the thing I'm trying to estimate, and B is the abstraction of a new observation or measurement.
Think of it this way: I have some knowledge of my prior estimate, and I'm able to characterize it as a Normally distributed random variable. It has a mean and a covariance in 3D space (let's call those x and P). There is a multivariate function for that pdf (lowercase p); it's an exponential of the negative square of the Mahalanobis distance normalized to have a unit volume. The same thing applies for my measurement, except it's only a 2D mean and covariance (let's call those z and R). I already know that I can use my H matrix, some matrix inverses and multiplications to combine this information. It's really just a case of multiplying the pdf for the state estimate and the measurement and re-normalizing to a unit volume. In other words, I believe both things are true, and they are independent, so I can multiply their pdfs and re-normalize to get a combined result. The rest is just linear algebra.
Now let's get to Bayes. P(A) is supposed to be the probability of an "event" or predicate. I can squint sideways and translate that as integrate my pdf for the prior estimate (Normal with mean x and covariance P) over a some unspecified bounds and convert that to a probability of my estimate being in those bounds, but I'm not sure that's what is intended. Again a similar thing applies for B (Normal with mean z and covariance R). It's frustrating that the bounds are never stated, because they could be radically different spaces in the numerator and denominator. I guess I'll give that a pass because maybe the notation would be too cumbersome if it was included, but this simplification seems to never be stated in any of the books or papers I've read on it.
Next, P(B|A) has to be probability of another "event", and if I squint again, the best I'm able to come up with is that it means a new Normal pdf with mean = H'z and covariance = H'RH. However, I haven't seen that spelled out any where, and so really that's just assuming the conclusion I want, which is super questionable. It also doesn't help me understand the left side of the Bayes equation - that vertical bar there seems to mean something different. When I look to the definition of conditional probability, I don't see anything about the vertical bar applied to multivariate Normal distributions.
If you translate it all to English, it reads as a coherent sentence, and that's fine. This is the "prior state estimate", that's the "observation", and this other thing is that's the "a posteriori" etc... However, if Bayes really helps with the understanding the math, the vertical bar has to mean something specific as an operator, and one would hope it meant the same thing on the left and right sides of the equation.
Similarly, in order for all of those P( ... ) to become scalars so multiplication and division are well defined, you need some integration bounds to turn the event into a probability. But at this point, I'm not even sure if that's what is intended by the notation - maybe they aren't scalars, and multiplication and division mean something radically different here. I honestly don't know.