Adaptive priors

Question

A lot of recent literature in Bayesian approach to inverse problems involves Adaptive priors, i.e - priors that depend on noise level. A lot of articles deal with optimization of contraction rates using Adaptive priors. My understanding is that in Bayesian statistics, the prior is constructed from information about the parameter space beforehand. If that is the case, what is the justification behind using adaptive priors. Any explanations and references would be most welcome.

Answer 1

The modification of the traditional Bayes method where the prior distribution is determined by the data goes by the name empirical Bayes method. The motivation for the empirical approach is a practical one: Not always is there enough information a priori available to meaningfully obtain the prior distribution. The justification for the empirical approach is that for large data sets, it leads to the same inferential conclusions as the traditional approach. For a comparison of the two methods see On Convergence Rates of Empirical Bayes Procedures.