  Approximate Bayesian Computation (ABC) has become one of the major tools of
likelihood-free statistical inference in complex mathematical models.
Simultaneously, stochastic differential equations (SDEs) have developed to an
established tool for modelling time dependent, real world phenomena with
underlying random effects. When applying ABC to stochastic models, two major
difficulties arise. First, the derivation of effective summary statistics and
proper distances is particularly challenging, since simulations from the
stochastic process under the same parameter configuration result in different
trajectories. Second, exact simulation schemes to generate trajectories from
the stochastic model are rarely available, requiring the derivation of suitable
numerical methods for the synthetic data generation. To obtain summaries that
are less sensitive to the intrinsic stochasticity of the model, we propose to
build up the statistical method (e.g., the choice of the summary statistics) on
the underlying structural properties of the model. Here, we focus on the
existence of an invariant measure and we map the data to their estimated
invariant density and invariant spectral density. Then, to ensure that these
model properties are kept in the synthetic data generation, we adopt
measure-preserving numerical splitting schemes. The derived property-based and
measure-preserving ABC method is illustrated on the broad class of partially
observed Hamiltonian type SDEs, both with simulated data and with real
electroencephalography (EEG) data. The proposed ingredients can be incorporated
into any type of ABC algorithm and directly applied to all SDEs that are
characterised by an invariant distribution and for which a measure-preserving
numerical method can be derived.
