  Maximum Likelihood Estimators (MLE) has many good properties. For example,
the asymptotic variance of MLE solution attains equality of the asymptotic
Cram{\'e}r-Rao lower bound (efficiency bound), which is the minimum possible
variance for an unbiased estimator. However, obtaining such MLE solution
requires calculating the likelihood function which may not be tractable due to
the normalization term of the density model. In this paper, we derive a
Discriminative Likelihood Estimator (DLE) from the Kullback-Leibler divergence
minimization criterion implemented via density ratio estimation and a Stein
operator. We study the problem of model inference using DLE. We prove its
consistency and show that the asymptotic variance of its solution can attain
the equality of the efficiency bound under mild regularity conditions. We also
propose a dual formulation of DLE which can be easily optimized. Numerical
studies validate our asymptotic theorems and we give an example where DLE
successfully estimates an intractable model constructed using a pre-trained
deep neural network.
