  Models of stochastic processes are widely used in almost all fields of
science. Theory validation, parameter estimation, and prediction all require
model calibration and statistical inference using data. However, data are
almost always incomplete observations of reality. This leads to a great
challenge for statistical inference because the likelihood function will be
intractable for almost all partially observed stochastic processes. This
renders many statistical methods, especially within a Bayesian framework,
impossible to implement. Therefore, computationally expensive likelihood-free
approaches are applied that replace likelihood evaluations with realisations of
the model and observation process. For accurate inference, however,
likelihood-free techniques may require millions of expensive stochastic
simulations. To address this challenge, we develop a new method based on recent
advances in multilevel and multifidelity. Our approach combines the multilevel
Monte Carlo telescoping summation, applied to a sequence of approximate
Bayesian posterior targets, with a multifidelity rejection sampler to minimise
the number of computationally expensive exact simulations required for accurate
inference. We present the derivation of our new algorithm for likelihood-free
Bayesian inference, discuss practical implementation details, and demonstrate
substantial performance improvements. Using examples from systems biology, we
demonstrate improvements of more than two orders of magnitude over standard
rejection sampling techniques. Our approach is generally applicable to
accelerate other sampling schemes, such as sequential Monte Carlo, to enable
feasible Bayesian analysis for realistic practical applications in physics,
chemistry, biology, epidemiology, ecology and economics.
