Graduate Student Seminar: Jonathan Williams
Jonathan P. Williams
Non-penalized variable selection in high-dimensional linear model settings
via generalized fiducial inference
This is joint work with Jan Hannig
Standard penalized methods of variable selection and parameter estimation rely on the magnitude of coefficient estimates to decide which variables to include in the final model. However, coefficient estimates are unreliable when the design matrix is collinear. To overcome this challenge an entirely new method of variable selection is presented within a generalized fiducial inference framework. This new procedure is able to effectively account for linear dependencies among subsets of covariates in a high-dimensional setting where p can grow almost exponentially in n.
It is shown that the procedure very naturally assigns small probabilities to subsets of covariates which include redundancies by way of explicit L0 minimization. Furthermore, with a typical sparsity assumption, it is shown that the proposed method is consistent in the sense that the probability of the true sparse subset of covariates converges in probability to 1 as n → ∞ and p → ∞. Very reasonable conditions are needed, and little restriction is placed on the class of 2p possible subsets of covariates to achieve this consistency result.