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This paper obtains convergence bounds for the concentration of Bayesian posterior distributions (around the true distribution) using a novel method that simplifies and enhances previous results. Based on the analysis, one also introduces a generalized family of Bayesian posteriors, and show that the convergence behavior of these generalized posteriors is completely determined by the local prior structure around the true distribution. This important and surprising robustness property does not hold for the standard Bayesian posterior in that it may not concentrate when there exist "Bad" prior structures even at places far away from the true distribution.
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