A Parzen-Based Distance Between Probability Measures as an Alternative of Summary Statistics in Approximate Bayesian Computation

Approximate Bayesian Computation (ABC) are likelihood-free Monte Carlo methods. ABC methods use a comparison between simulated data, using different parameters drawn 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 a sufficiently smooth 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.