In this article we focus on Maximum Likelihood estimation (MLE) for the static model parameters of hidden Markov models (HMMs). We will consider the case where one cannot or does not want to compute the conditional likelihood
density of the observation given the hidden state because of increased computational
complexity or analytical intractability. Instead we will assume that one may obtain
samples from this conditional likelihood and hence use approximate Bayesian
computation (ABC) approximations of the original HMM. Although these ABC
approximations will induce a bias, this can be controlled to arbitrary precision via
a positive parameter , so that the bias decreases with decreasing . We first establish
that when using an ABC approximation of the HMM for a fixed batch of data,
then the bias of the resulting log- marginal likelihood and its gradient is no worse
than O(n), where n is the total number of data-points. Therefore, when using
gradient methods to perform MLE for the ABC approximation of the HMM, one
may expect parameter estimates of reasonable accuracy. To compute an estimate of
the unknown and fixed model parameters, we propose a gradient approach based on
simultaneous perturbation stochastic approximation (SPSA) and Sequential Monte
Carlo (SMC) for the ABC approximation of the HMM. The performance of this
method is illustrated using two numerical examples.