  Approximate Bayesian Computation (ABC) are likelihood-free Monte Carlo
methods. ABC methods use a comparison between simulated data, using different
parameters drew from a prior distribution, and observed data. This comparison
process is based on computing a distance between the summary statistics from
the simulated data and the observed data. For complex models, it is usually
difficult to define a methodology for choosing or constructing the summary
statistics. Recently, a nonparametric ABC has been proposed, that uses a
dissimilarity measure between discrete distributions based on empirical kernel
embeddings as an alternative for summary statistics. The nonparametric ABC
outperforms other methods including ABC, kernel ABC or synthetic likelihood
ABC. However, it assumes that the probability distributions are discrete, and
it is not robust when dealing with few observations. In this paper, we propose
to apply kernel embeddings using an smoother density estimator or Parzen
estimator for comparing the empirical data distributions, and computing the ABC
posterior. Synthetic data and real data were used to test the Bayesian
inference of our method. We compare our method with respect to state-of-the-art
methods, and demonstrate that our method is a robust estimator of the posterior
distribution in terms of the number of observations.
