  We research relations between optimal transport theory (OTT) and approximate
Bayesian computation (ABC) possibly connected to relevant metrics defined on
probability measures. Those of ABC are computational methods based on Bayesian
statistics and applicable to a given generative model to estimate its a
posteriori distribution in case the likelihood function is intractable. The
idea is therefore to simulate sets of synthetic data from the model with
respect to assigned parameters and, rather than comparing prospects of these
data with the corresponding observed values as typically ABC requires, to
employ just a distance between a chosen distribution associated to the
synthetic data and another of the observed values. Our focus lies in
theoretical and methodological aspects, although there would exist a remarkable
part of algorithmic implementation, and more precisely issues regarding
mathematical foundation and asymptotic properties are carefully analysed,
inspired by an in-depth study of what is then our main bibliographic reference,
that is Bernton et al. (2019), carrying out what follows: a rigorous
formulation of the set-up for the ABC rejection algorithm, also to regain a
transparent and general result of convergence as the ABC threshold goes to zero
whereas the number n of samples from the prior stays fixed; general technical
proposals about distances leaning on OTT; weak assumptions which lead to lower
bounds for small values of threshold and as n goes to infinity, ultimately
showing a reasonable possibility of lack of concentration which is contrary to
what is proposed in Bernton et al. (2019) itself.
