  Penalized likelihood models are widely used to simultaneously select
variables and estimate model parameters. However, the existence of weak signals
can lead to inaccurate variable selection, biased parameter estimation, and
invalid inference. Thus, identifying weak signals accurately and making valid
inferences are crucial in penalized likelihood models. We develop a unified
approach to identify weak signals and make inferences in penalized likelihood
models, including the special case when the responses are categorical. To
identify weak signals, we use the estimated selection probability of each
covariate as a measure of the signal strength and formulate a signal
identification criterion. To construct confidence intervals, we propose a
two-step inference procedure. Extensive simulation studies show that the
proposed procedure outperforms several existing methods. We illustrate the
proposed method by applying it to the Practice Fusion diabetes data set.
