Bayesian model selection. What are its frequentist guarantees?
Bayesian model selection can be justified from a purely subjective point of view, given bona fide priors Bayes theorem gives posterior model probabilities. These can be used to select the best (in some sense) model and assess how certain we are about that choice. An important question however is how good are these procedures when viewed from a frequentist viewpoint. As one obtains more data, can we recover the optimal solution, how quickly, and what happens when the data-generating truth does not lie in the considered model family? As one pushes for sparsity, what is the loss in power to detect non-spurious signals? Such questions are more pressing in high dimensions where the number of models grows with n. We review some standard and recent results. Specifically, we will present a novel framework to study posterior concentration. The framework is fully general and helps hightlight the main intuitive principles, which we shall illustrate in canonical high-dimensional linear regression.
Palabras clave / Keywords: model selection posterior concentration sensitivity-sparsity trade-off
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