  Matrix completion has become an extremely important technique as data
scientists are routinely faced with large, incomplete datasets on which they
wish to perform statistical inferences. We investigate how error introduced via
matrix completion affects statistical inference. Furthermore, we prove recovery
error bounds which depend upon the matrix recovery error for several common
statistical inferences. We consider matrix recovery via nuclear norm
minimization and a variant, $\ell_1$-regularized nuclear norm minimization for
data with a structured sampling pattern. Finally, we run a series of numerical
experiments on synthetic data and real patient surveys from MyLymeData, which
illustrate the relationship between inference recovery error and matrix
recovery error. These results indicate that exact matrix recovery is often not
necessary to achieve small inference recovery error.
