Title: Bayes model selection,
Abstract: We offer a general Bayes theoretic framework to tackle the model selection
problem under a two-step prior design: the first-step prior serves to assess
the model selection uncertainty, and the second-step prior quantifies the prior
belief on the strength of the signals within the model chosen from the first
step.
We establish non-asymptotic oracle posterior contraction rates under (i) a
new Bernstein-inequality condition on the log likelihood ratio of the
statistical experiment, (ii) a local entropy condition on the dimensionality of
the models, and (iii) a sufficient mass condition on the second-step prior near
the best approximating signal for each model. The first-step prior can be
designed generically. The resulting posterior mean also satisfies an oracle
inequality, thus automatically serving as an adaptive point estimator in a
frequentist sense. Model mis-specification is allowed in these oracle rates.
The new Bernstein-inequality condition not only eliminates the convention of
constructing explicit tests with exponentially small type I and II errors, but
also suggests the intrinsic metric to use in a given statistical experiment,
both as a loss function and as an entropy measurement. This gives a unified
reduction scheme for many experiments considered in Ghoshal & van der
Vaart(2007) and beyond. As an illustration for the scope of our general results
in concrete applications, we consider (i) trace regression, (ii)
shape-restricted isotonic/convex regression, (iii) high-dimensional partially
linear regression and (iv) covariance matrix estimation in the sparse factor
model. These new results serve either as theoretical justification of practical
prior proposals in the literature, or as an illustration of the generic
construction scheme of a (nearly) minimax adaptive estimator for a
multi-structured experiment.