Sometimes you can’t use Bayesian methods literally; often, indeed. But when you can use the exact Bayesian calculation that uses every scrap of available knowledge, you are done. You will never find a statistical method that yields a better answer. You may find a cheap approximation that works excellently nearly all the time, and it will be cheaper, but it will not be more accurate. Not unless the other method uses knowledge, perhaps in the form of disguised prior information, that you are not allowing into the Bayesian calculation; and then when you feed the prior information into the
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