We derive a generalized likelihood ratio method to estimate any distribution sensitivity in a unified form. Then, the distribution sensitivities are used to in a gradient-based simulated maximum likelihood estimation (GSMLE) to estimate unknown parameters in a stochastic model without assuming that the likelihoods of the observations are available in closed form. GSMLE can direct fit the underlying stochastic model to the output data, which opens the possibility of extending data-driven ideas to complex (causal) stochastic models. In addition, GSMLE can efficiently address the calibration of hidden Markov model, which has been considered as a difficult problem in statistics and econometrics.
