MC 20: Workshop on Numerical Methods for Stochastic Differential Equations
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15:15: Erik Lindström, Centre for the Mathematical Sciences, Lund University: Introduction
15:30: Irene Tubikanec, Johannes Kepler University Linz, Austria: Spectral Density-Based and Measure-Preserving ABC for Partially Observed SDEs With an Invariant Distribution
16:15: Samuel Wiqvist, Centre for the Mathematical Sciences, Lund University: Efficient inference for stochastic differential equation mixedeffects models using correlated particle pseudo-marginal algorithms
Let us consider a n-dimensional SDE whose solution process is only partially observed through a one-dimensional and parameter-dependent output process admitting an invariant distribution. We aim to infer the parameters from discrete time measurements of the invariant output process. Due to the increased model complexity, needed to understand and reproduce the real data, the underlying likelihood is often unknown or intractable. Among several likelihood-free inference methods, we focus on the Approximate Bayesian Computation (ABC) approach .
When applying ABC to stochastic processes, two 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 for general SDEs, and thus a numerical method for the synthetic data generation within the ABC
framework has to be derived. We tackle these issues as follows. To reduce the randomness coming from the underlying model, we propose to make use of the structural properties of the SDE, namely the existence of a unique invariant distribution. We map the synthetic data into their estimated invariant density and invariant spectral density, almost eliminating the variability in the data, and thus making hidden information about the parameters accessible. Since our ABC algorithm is based on the structural property, it can only lead to successful inference when the invariant measure is preserved in the synthetic data simulation. To achieve this, we propose to use a structure-preserving numerical scheme that, differently from the commonly used Euler-Maruyama method, preserves the properties of the underlying SDE.
Here, we illustrate our proposed Spectral Density-Based and Measure-Preserving ABC Algorithm  on the stochastic Jansen and Rit Neural Mass Model (JR-NMM) . This model is ergodic, which results in the fact that the output process admits an invariant distribution exponentially fast. The use of numerical splitting schemes guarantees the preservation of the invariant distribution in the synthetic data generation step. With our new approach, we succeed in the simultaneous estimation of the three most crucial parameters of the JR-NMM. Finally, we apply our method to fit the JR-NMM to real EEG alpha-rhythmic recordings
 M.A. Beaumont, W. Zhang, D.J. Balding ”Approximate Bayesian computation in population genetics.” In: Genetics, 162(4), pp.2025-2035 (2002)
 E. Buckwar, M. Tamborrino, I. Tubikanec. ”Spectral Density-Based and Measure-Preserving ABC for Partially Observed Diffusion Processes. An Illustration on Hamiltonian SDEs.” preprint available at arXiv:1903.01138
 M. Ableidinger, E. Buckwar, H. Hinterleitner. ”A Stochastic Version of the Jansen and Rit Neural Mass Model: Analysis and Numerics.” In: The Journal of Mathematical Neuroscience 7(8) (2017)
Stochastic differential equation mixed-effects models (SDEMEMs) are flexible hierarchical models that are able to account for random variability inherent in the underlying time-dynamics, as well as the variability between experimental units and, optionally, account for measurement error. We consider inference for state-space SDEMEMs, however the inference problem is complicated by the typical intractability of the observed data likelihood which motivates the use of sampling-based approaches such as Markov chain Monte Carlo. Our proposed approach is the use of a Gibbs sampler to target the marginal posterior of all parameter values of interest. Our algorithm is made computationally efficient through careful use of blocking strategies and correlated pseudo-marginal Metropolis-Hastings steps within the Gibbs scheme. The resulting methodology is flexible and is able to deal with a large class of SDEMEMs. We demonstrate the methodology on state-space models describing two applications. For these two applications, we found that our algorithm is about ten to forty times more efficient, depending on the considered application, than similar algorithms not using correlated particle filters. For the second application we will also compare ordinary differential equation mixed-effects models (ODEMEMs) with SDEMEMs.
Joint work with: Andrew Golightly, Ashleigh T. Mclean, and Umberto Picchini.
Link to paper: https://arxiv.org/pdf/1907.09851.pdf