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In frequentist statistics, one can hardly take a sip of coffee without someone mentioning Fisher information and the Cramer-Rao lower bound. On the other hand, from my limited experience in the field of statistics, the analogous van Trees inequality (sometimes referred to as the Bayesian Cramer-Rao lower bound) isn't mentioned nearly as much.

Is this just my (mis)perception? Or is there some good reason why Bayesians don't care as much about such inequalities? Perhaps it is simply because it is harder to compute?

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I would wager that this has a lot to do with the fact that many Bayesian models do not have tractable posterior distributions so that the van Trees inquality is likely to be of little use. Additionally, a major thrust of Bayesian research over the last 30 years or so has been on Markov chain Monte Carlo (MCMC) methods. MCMC methods are a collection of methods to draw samples from the posterior distribution. With enough of these samples you can calculate the numeric value of any variances you want and do not need to worry about using the van Trees inequality.

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