Undirected graphs are often used to describe high dimensional distributions.
Under sparsity conditions, the graph can be estimated using
{\ell}1-penalization methods. We propose and study the following method. We
combine a multiple regression approach with ideas of thresholding and
refitting: first we infer a sparse undirected graphical model structure via
thresholding of each among many {\ell}1-norm penalized regression functions; we
then estimate the covariance matrix and its inverse using the maximum
likelihood estimator.