  Approximate Bayesian Computation (ABC) methods are commonly used to
approximate posterior distributions in models with unknown or computationally
intractable likelihoods. Classical ABC methods are based on nearest neighbor
type algorithms and rely on the choice of so-called summary statistics,
distances between datasets and a tolerance threshold. Recently, methods
combining ABC with more complex machine learning algorithms have been proposed
to mitigate the impact of these "user-choices". In this paper, we propose the
first, to our knowledge, ABC method completely free of summary statistics,
distance and tolerance threshold. Moreover, in contrast with usual
generalizations of the ABC method, it associates a confidence interval (having
a proper frequentist marginal coverage) with the posterior mean estimation (or
other moment-type estimates).
  Our method, ABCD-Conformal, uses a neural network with Monte Carlo Dropout to
provide an estimation of the posterior mean (or others moment type functional),
and conformal theory to obtain associated confidence sets. Efficient for
estimating multidimensional parameters, we test this new method on three
different applications and compare it with other ABC methods in the literature.
