  Penalized regression methods, most notably the lasso, are a popular approach
to analyzing high-dimensional data. An attractive property of the lasso is that
it naturally performs variable selection. An important area of concern,
however, is the reliability of these variable selections. Motivated by local
false discovery rate methodology from the large-scale hypothesis testing
literature, we propose a method for calculating a local false discovery rate
for each variable under consideration by the lasso model. These rates can be
used to assess the reliability of an individual feature, or to estimate the
model's overall false discovery rate. The method can be used for all values of
$\lambda$. This is particularly useful for models with a few highly significant
features but a high overall Fdr, which are a relatively common occurrence when
using cross validation to select $\lambda$. It is also flexible enough to be
applied to many varieties of penalized likelihoods including GLM and Cox
models, and a variety of penalties, including MCP and SCAD. We demonstrate the
validity of this approach and contrast it with other inferential methods for
penalized regression as well as with local false discovery rates for univariate
hypothesis tests. Finally, we show the practical utility of our method by
applying it to two case studies involving high dimensional genetic data.
