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Not sure what you're asking. Kalman filters are a standard part of control toolkits, and they're "optimal" in the sense of being the optimal LQG solution for linear time invariant systems with gaussian noise. They rest on something called the "certainty equivalence principle"
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Thanks, yeah, I was being vague. Kalman filters are optimal for the thing they are optimal for, and then there’s various broader classes of problems for which analogous things are provably optimal (?), and you’d like a strong result for a very broad class that isn’t done…
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That is exactly correct. Under some nice conditions, a noisy signal can be used in place of a non-noisy signal and you'll still get an optimal solution. LQG is a weird but practically important special case that reduces to a simpler case.
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To your larger point re: expressivity of logics, control theory can be viewed as a "hack" where we work with rare islands of tractability where probabilistic systems behave as simply as first-order predicate logic systems despite not being generally reduiable to them
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