  Approximate Bayesian computation (ABC) is a class of Bayesian inference
algorithms that targets for problems with intractable or {unavailable}
likelihood function. It uses synthetic data drawn from the simulation model to
approximate the posterior distribution. However, ABC is computationally
intensive for complex models in which simulating synthetic data is very
expensive. In this article, we propose an early rejection Markov chain Monte
Carlo (ejMCMC) sampler based on Gaussian processes to accelerate inference
speed. We early reject samples in the first stage of the kernel using a
discrepancy model, in which the discrepancy between the simulated and observed
data is modeled by Gaussian process (GP). Hence, the synthetic data is
generated only if the parameter space is worth exploring. We demonstrate from
theory, simulation experiments, and real data analysis that the new algorithm
significantly improves inference efficiency compared to existing
early-rejection MCMC algorithms. In addition, we employ our proposed method
within an ABC sequential Monte Carlo (SMC) sampler. In our numerical
experiments, we use examples of ordinary differential equations, stochastic
differential equations, and delay differential equations to demonstrate the
effectiveness of the proposed algorithm. We develop an R package that is
available at https://github.com/caofff/ejMCMC.
