relationship and differences between Bayesian methods

Shrinkage parameter estimation:

(1) Ridge regression: grid search, or a mixed model approach (with the "emma" R package)

(2) Bayesian ridge regression: a Bayesian hierarchical model

(1) and (2) assume all markers have a common variance and therefore shrink equally for each marker effect

(3) LASSO: a descent search (with the "glmnet" package), elastic net: a grid search via cross-validation for the shrinkage parameter controlling the relative amount of L1 and L2 penalties. The model with the minimum MSE was selected.

(4) Bayesian LASSO: 

Differences:

(1) The BL produces stronger shrinkage of regression coefficients that are close to zero and less shrinkage of those with large absolute value, leading to a sparse model. Whereas RR-BLUP shrinks more strongly the regression coefficients with a large value.

(2) The BL does not select variables by assigning coefficients to 0 as does LASSO (non-Bayesian version)