Abstract
We propose to accelerate Hamiltonian and Lagrangian Monte Carlo algorithms by coupling them with Gaussian processes for emulation of the log unnormalised posterior distribution. We provide proofs of detailed balance with respect to the exact posterior distribution for these algorithms, and validate the correctness of the samplers’ implementation by Geweke consistency tests. We implement these algorithms in a delayed acceptance (DA) framework, and investigate whether the DA scheme can offer computational gains over the standard algorithms. A comparative evaluation study is carried out to assess the performance of the methods on a series of models described by differential equations, including a real-world application of a 1D fluid-dynamics model of the pulmonary blood circulation. The aim is to identify the algorithm which gives the best trade-off between accuracy and computational efficiency, to be used in nonlinear DE models, which are computationally onerous due to repeated numerical integrations in a Bayesian analysis. Results showed no advantage of the DA scheme over the standard algorithms with respect to several efficiency measures based on the effective sample size for most methods and DE models considered. These gradient-driven algorithms register a high acceptance rate, thus the number of expensive forward model evaluations is not significantly reduced by the first emulator-based stage of DA. Additionally, the Lagrangian Dynamical Monte Carlo and Riemann Manifold Hamiltonian Monte Carlo tended to register the highest efficiency (in terms of effective sample size normalised by the number of forward model evaluations), followed by the Hamiltonian Monte Carlo, and the No U-turn sampler tended to be the least efficient.
Highlights
Parameter estimation and uncertainty quantification (UQ) in systems of nonlinear ordinary and partial differential equations (ODEs/PDEs) is a topical research area with the emergence of complex mathematical models expressed via ODEs or PDEs
We have provided theoretical and empirical investigations into Hamiltonian/Lagrangian Monte Carlo algorithms coupled with Gaussian processes for emulation of the log unnormalised posterior distribution
We have proved that these emulation algorithms satisfy detailed balance with respect to the exact posterior distribution
Summary
Parameter estimation and uncertainty quantification (UQ) in systems of nonlinear ordinary and partial differential equations (ODEs/PDEs) is a topical research area with the emergence of complex mathematical models expressed via ODEs or PDEs. Statistical inference allows estimation of the unknown model parameters from the data in a robust and coherent manner within a Bayesian or frequentist framework. This is a challenging task to accomplish since nonlinear ODE/PDE models that faithfully capture real-world processes of interest are analytically intractable and can only be solved using numerical integration. LDMC (Lan et al 2015) simulates Lagrangian dynamics (instead of Hamiltonian dynamics), the ‘velocity’ variables replace the ‘momentum’ variables This enables the use of an explicit geometric integrator, which substantially improves the computational efficiency of the costly RMHMC implicit integrator. In LDMC, to RMHMC, M is adjusted to the curvature of the (approximate) posterior distribution at every step throughout the trajectory
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