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Spectral Density-Based and Measure-Preserving ABC for partially observed diffusion processes. An illustration on Hamiltonian SDEs

by   Evelyn Buckwar, et al.

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 processes, two major difficulties arise. First, different realisations from the output process with the same choice of parameters may show a large variability due to the stochasticity of the model. Second, exact simulation schemes are rarely available, requiring the derivation of suitable numerical methods for the synthetic data generation. To reduce the randomness in the data coming from the SDE, 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 the model properties are kept in the synthetic data generation, we adopt a structure-preserving numerical scheme. The derived property-based and measure-preserving ABC method is illustrated on the broad class of partially observed Hamiltonian SDEs, both with simulated data and with real electroencephalography (EEG) data. The proposed ABC method can be directly applied to all SDEs that are characterised by an invariant distribution and for which a measure-preserving numerical method can be derived.


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