In Kalman filter, we can show it's a minimum variance filter, which I believe is due to the linearity of system and the Gaussianity of noise. It comes to me that what is the optimality criterion used in Bayesian filtering, from which we can derive Kalman filter. I usually treated the optimality of Bayesian filter as granted because it's based on Bayes theorem, which in turn is the foundation of Bayesian statistics. However, I'm not sure if we can have a more concrete metric, such as mean square error or other metrics, that can be used to show that Bayesian filtering is optimizing such a metric.
I searched online and found the paper "Bayesian Filtering: From Kalman Filters to Particle Filters, and Beyond ". However, I'm confused about the following statement from the paper:
The first part of the sentence says MMSE (minimum mean squared error) is the optimality criterion used for Bayesian filtering. But I cannot see how the posterior distribution from Bayesian filtering minimizes a mean squared error. So I feel this sentence says if we want a point estimate, we obtain MMSE (which is the conditional mean) from the posterior distribution obtained by Bayesian filtering. Is this correct understanding?
The second part of the sentence seems quite vague to me. It seems imply that, since we are using Bayes theorem to obtain the posterior distribution, we can feel assured that we are doing things in an optimal way. Does this sound correct?