Bayesian prior update feedback

Question

Consider a standard Bayesian MCMC inference problem with $\theta_n$ free parameters. I know very little about their distributions, so I solve using uniform priors. Then I take, for example, the mean and standard deviation of their estimated distributions, and use them as Gaussian priors in a new run. I keep doing this until some stopping condition.

I'm certain I'm not the first to come up with this idea, but I can't find the "proper" name for this process. It almost feels like I'm "bootstrapping" my Bayesian setup.

Can (should) this be done?

Answer 1

In 1993, I called this method prior feedback. In 1998, we used it for hidden Markov models with Mike Titterington. In 2002, we also exploited this idea in a Statistics and Computing paper with Arnaud Doucet et Simon Godsill under the acronym of SAME (state augmentation for marginal estimation). The later was "reinvented" by several authors like Lele's 2007 data cloning, Gaetan & Yao's 2003 multiple imputation Metropolis, and Johannes & Polson's 2007 MCMC maximum likelihood, The approach converges to a point mass at the maximum likelihood estimator, provided it stands within the support of the prior. It is much more "annealing" (since this almost amounts to spiking the likelihood to an ever increasing power) than "bootstrapping", although increasing the bootstrap sample would eventually come to the MLE as well.