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Title: Estimation of the Derivatives of Functions that Can Only be Evaluated with Noise
Abstract: Iterated Filtering methods have recently been introduced to perform maximum likelihood parameter estimation in state-space models, and they only require being able to simulate the latent Markov model according to its prior distribution. They rely on an approximation of the score vector for general statistical models based upon an artificial posterior distribution and bypasses the calculation of any derivative. We show here that this score estimator can be derived from a simple application of Stein’s lemma and how an additional application of this lemma provides an original derivative-free estimator of the observed information matrix. These methods tackle the general problem of estimating the first two derivatives of a function that can only be evaluated point-wise with some noise. We compare these new methods with finite difference schemes and make connections with proximal mappings. In particular we look at the bias and variance of these estimators, the effect of the variance of the noise, and the effect of the dimension of the parameter space.