Abstract
Ionides, King et al. (see e.g. Inference for nonlinear dynamical systems,
PNAS 103) have recently introduced an original approach to perform maximum
likelihood parameter estimation in state-space models which only requires being
able to simulate the latent Markov model according to its prior distribution.
Their methodology relies 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. We establish that these estimators exhibit
robustness properties compared to finite difference estimators while their bias
and variance scale as well as finite difference type estimators, including
simultaneous perturbations (see e.g. Spall, IEEE Trans. on Automatic Control
37), with respect to the dimension of the parameter. For state-space models
where sequential Monte Carlo computation is required, these estimators can be
further improved. In this specific context, we derive original derivative-free
estimators of the score vector and observed information matrix which are
computed using sequential Monte Carlo approximations of smoothed additive
functionals associated with a modified version of the original state-space
model.
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