GpABC

GpABC provides algorithms for likelihood - free parameter inference and model selection using Approximate Bayesian Computation (ABC). Two sets of algorithms are available:

• Simulation based - full simulations of the model(s) is done on each step of ABC.
• Emulation based - a small number of simulations can be used to train a regression model (the emulator), which is then used to approximate model simulation results during ABC.

GpABC offers Gaussian Process Regression (GPR) as an emulator, but custom emulators can also be used. GPR can also be used standalone, for any regression task.

Stochastic models, that don't conform to Gaussian Process Prior assumption, are supported via Linear Noise Approximation (LNA).

Installation

GpABC can be installed using the Julia package manager. From the Julia REPL, type ] to enter the Pkg REPL mode and run

pkg> add GpABC

Notation

In parts of this manual that deal with Gaussian Processes and kernels, we denote the number of training points as $n$, and the number of test points as $m$. The number of dimensions is denoted as $d$.

In the context of ABC, vectors in parameter space ($\theta$) are referred to as particles. Particles that are used for training the emulator (training_x) are called design points. To generate the distances for training the emulator (training_y), the model must be simulated for the design points.

Examples

• ABC parameter estimation example
• ABC model selection example
• Stochastic Inference (LNA) example
• Gaussian Process regression example

Dependencies

• Optim - for training Gaussian Process hyperparameters.
• Distributions - probability distributions.
• Distances - distance functions
• OrdinaryDiffEq - for solving ODEs for LNA, and also used throughout the examples for model simulation (ODEs and SDEs)
• ForwardDiff - automatic differentiation is also used by LNA

References

• Toni, T., Welch, D., Strelkowa, N., Ipsen, A., & Stumpf, M. P. H. (2009). Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems. Interface, (July 2008), 187–202. https://doi.org/10.1098/rsif.2008.0172
• Filippi, S., Barnes, C. P., Cornebise, J., & Stumpf, M. P. H. (2013). On optimality of kernels for approximate Bayesian computation using sequential Monte Carlo. Statistical Applications in Genetics and Molecular Biology, 12(1), 87–107. https://doi.org/10.1515/sagmb-2012-0069
• Rasmussen, C. E., & Williams, C. K. I. (2006). Gaussian Processes for Machine Learning. MIT Press. ISBN 0-262-18253-X. http://www.gaussianprocess.org/gpml
• Schnoerr, D., Sanguinetti, G., & Grima, R. (2017). Approximation and inference methods for stochastic biochemical kinetics—a tutorial review. Journal of Physics A: Mathematical and Theoretical, 50(9), 093001. https://doi.org/10.1088/1751-8121/aa54d9
• Karlebach, G., & Shamir, R. (2008). Modelling and analysis of gene regulatory networks. Nature Reviews Molecular Cell Biology, 9(10), 770–780. https://doi.org/10.1038/nrm2503