Your TLDR is slightly wrong. The Kalman filter is exactly Bayes theorem for linear dynamics and additive Gaussian noise. Bayes theorem is more general, but thes

Your TLDR is slightly wrong. The Kalman filter is exactly Bayes theorem for linear dynamics and additive Gaussian noise. Bayes theorem is more general, but these assumptions yield a simple set of linear equations that can be solved very quickly. In general, using Bayes theorem for filtering is often called the Bayes filter. Outside of a limited set of circumstances, it is not possible to implement. The Kalman filter is one set of circumstances because linear transforms of Gaussian distributions remain Gaussian, and we can specify a Gaussian process just by its mean and covariance. Another set of circumstances is when you have a discrete system with a small enough number of states. In general however, the Bayes filter results in an infinite dimensional dynamical system since it yields a typically continuous function representing the distribution of your state given your measurements. The Kalman filter, particle filter, and other methods approximate this problem via a more tractable finite dimensional representation.