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Safe Learning: How to Modify Bayesian Inference when All Models are Wrong

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Standard Bayesian inference can behave suboptimally if the model under consideration is wrong: in some simple settings, the posterior may fail to concentrate even in the limit of infinite sample size. We introduce a test that can tell from the data whether we are in such a situation. If we are, we can adjust the learning rate (equivalently: make the prior lighter-tailed) in a data-dependent way. The resulting “safe” estimator continues to achieve good rates with wrong models. When applied to classification, the approach achieves optimal rates under Tsybakov’s conditions, thereby creating a bridge between Bayes/MDL and statistical learning-style inference.

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