Approximate Bayesian computation with the Wasserstein distance

A growing range of generative statistical models prohibits the numerical evaluation of their likelihood functions. Approximate Bayesian computation has become a popular approach to overcome this issue, simulating synthetic data given parameters and comparing summaries of these simulations with the corresponding observed values. We propose to avoid these summaries and the ensuing loss of information through the use of Wasserstein distances between empirical distributions of observed and synthetic data. We describe how the approach can be used in the setting of dependent data such as time series, and how approximations of the Wasserstein distance allow in practice the method to scale to large datasets. In particular, we propose a new approximation to the optimal assignment problem using the Hilbert space-filling curve. The approach is illustrated on various examples including i.i.d. data and time series.