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jsteinhardt comments on Beyond Bayesians and Frequentists - Less Wrong
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I'm confused by what you are suggesting here. Even a Bayesian method making no approximations at all doesn't necessarily have guaranteed performance (see my response to Oscar_Cunningham).
I'm referring to using an approximation in order to guarantee performance. E.g. replacing the sum of a bunch of independent, well-behaved random variables with a gaussian, and using monte-carlo methods to get approximate properties of the individual random variables with known resources if necessary.