Spectral density-based and measure-preserving ABC for partially observed diffusion processes
Sprache des Vortragstitels:
Second Italian Meeting on Probability and Mathematical Statistics
Sprache des Tagungstitel:
Over the last decades, stochastic differential equations (SDEs) have become an established tool for modelling time dependent real world phenomena that underlie random effects. Besides the modelling, it is of primary interest to estimate the underlying model parameters. This is particularly difficult when the n-dimensional stochastic process is only partially observed through a function of its coordinates. Moreover, due to the increasing model complexity, the underlying likelihoods are often unknown or intractable. Among likelihood-free inference methods, here we focus on the Approximate Bayesian Computation (ABC) approach. 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 propose summary statistics that are less sensitive to the intrinsic stochasticity of the model, we propose to build up the statistical method 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 that, differently from the commonly used Euler-Maruyama method, preserves the properties of the underlying SDE. The derived Spectral Density-Based and Measure-Preserving ABC method is illustrated on the broad class of partially observed Hamiltonian SDEs, both with simulated and with real electroencephalography (EEG) data. The proposed ABC method can be directly applied to all SDEs characterised by an invariant distribution, for which a measure-preserving numerical scheme can be derived.