Spectral density-based and measure-preserving ABC for partially observed diffusion processes. An illustration on...

Area of Science:

• Computational Statistics
• Mathematical Modeling
• Time Series Analysis

Background:

• Approximate Bayesian computation (ABC) is crucial for likelihood-free inference in complex models.
• Stochastic differential equations (SDEs) model real-world phenomena with random effects.
• Applying ABC to SDEs faces challenges in summary statistics, distance metrics, and simulation schemes.

Purpose of the Study:

• To develop a robust ABC method for SDEs by leveraging underlying structural properties.
• To enhance the accuracy and reliability of statistical inference in complex stochastic models.
• To address the difficulties in summary statistics selection and synthetic data generation for SDEs.

Main Methods:

• Utilized invariant measure properties (invariant density and spectral density) for summary statistics.
• Employed measure-preserving numerical splitting schemes for synthetic data generation.
• Applied the property-based ABC method to Hamiltonian type SDEs and electroencephalography data.

Main Results:

• Derived summary statistics are robust to model stochasticity, improving ABC inference accuracy.
• The proposed measure-preserving ABC method significantly outperforms standard numerical methods like Euler-Maruyama.
• Demonstrated successful application on both simulated and real-world electroencephalography data.

Conclusions:

• The property-based and measure-preserving ABC approach offers accurate and reliable inference for SDEs.
• This method is adaptable to various ABC algorithms and applicable to SDEs with invariant distributions.
• The findings highlight the importance of structural model properties in likelihood-free inference.