estimation that uses approximate Bayesian computation (ABC, [@tavare1997inferring], [@pritchard1999population], [@marjoram2003markov]). The method is applicable to a very general experimental setup, valid for both “static” and “dynamic” models. Our main question is: since in ABC studies artificial datasets are produced from the data generating model and compared to the observed data according to a threshold parameter $\delta$ (the smaller the $\delta$ the better the inference), what can we do if we are unable to reduce $\delta$ below a certain level? Or alternatively, can we perform inference using a relatively large $\delta$ and fewer Markov chain Monte Carlo (MCMC) simulations, instead of progressively decrease $\delta$ at the expense of using many MCMC iterations? Suppose we have obtained a very rough approximation to the posterior distribution for the unknowns, if we have at least located the main mode for the posterior can we conduct approximate maximum likelihood inference? Basically by accepting of having obtained a poor approximation to the posterior, except for the location of its main mode, we switch to maximum likelihood estimation by proposing draws around such approximated mode using a special ABC-MCMC sampler.

Let $Y\sim f(y|\cdot,\phi)$ denote a realization from an observable random variable, i.e. $f(\cdot)$ is the data generating 