How to effectively select the predictors for Bayesian linear regression model? I have around 20 independent (or not strongly correlated) predictor variables. And there are about 20 observations for each variable and the outcome. 
I want to build a Bayesian linear regression model using these data. How to effectively select the predictors, since 20 predictors are too many?
Maybe can use information criteria (AIC, WAIC)? Do I have to try each possible combination of the model predictors?
 A: See Juho Piironen and Aki Vehtari (2017). Comparison of Bayesian predictive methods for model selection. Statistics and Computing, 27(3):711-735.
http://link.springer.com/article/10.1007/s11222-016-9649-y
which shows that most approaches overfit during the selection, and that projection predictive approach performs the best. 
The projection predictive model selection has been implemented in projpred R package https://github.com/stan-dev/projpred which supports rstanarm models.
For a quick intro see https://htmlpreview.github.io/?https://github.com/stan-dev/projpred/blob/master/vignettes/quickstart.html
We have tested projection predictive model selection with many small n large n datasets up to 20k predictors.
A: Several types of priors for conducting variable selection have been developed in the context of linear regression models. One the most recent proposals are non-local priors, which are implemented in the R package mombf:
https://cran.r-project.org/web/packages/mombf/index.html
In your case, $2^{20}$ models are not that many (1048576), and it may be feasible to explore all of them. If you want to conduct a faster, efficient selection, you may want to have a look at either of these two papers (which are also part of the mombf package):

Variable Selection Via Gibbs Sampling. Edward I. George; Robert E. McCulloch. Journal of the American Statistical Association

and

J.G. Scott and J.O Berger. [Bayes and empirical Bayes multiplicity adjustment in the variable selection problem](https://projecteuclid.org/euclid.aos/1278861454). The Annals of Statistics.

