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Background: Parameter estimation for differential equation models of intracellular processes is a highly relevant bu challenging task. The available experimental data do not usually contain enough information to identify all parameters uniquely, resulting in ill-posed estimation problems with often highly correlated parameters. Sampling-based Bayesian statistical approaches are appropriate for tackling this problem. The samples are typically generated via Markov chain Monte Carlo, however such methods are computationally expensive and their convergence may be slow, especially if there are strong correlations between parameters. Monte Carlo methods based on Euclidean or Riemannian Hamiltonian dynamics have been shown to outperform other samplers by making proposal moves that take the local sensitivities of the system's states into account and accepting these moves with high probability. However, the high computational cost involved with calculating the Hamiltonian trajectories prevents their widespread use for all but the smallest differential equation models. The further development of efficient sampling algorithms is therefore an important step towards improving the statistical analysis of predictive models of intracellular processes., Results: We show how state of the art Hamiltonian Monte Carlo methods may be significantly improved for steady state dynamical models. We present a novel approach for efficiently calculating the required geometric quantities by tracking steady states across the Hamiltonian trajectories using a Newton-Raphson method and employing local sensitivity information. Using our approach, we compare both Euclidean and Riemannian versions of Hamiltonian Monte Carlo on three models for intracellular processes with real data and demonstrate at least an order of magnitude improvement in the effective sampling speed. We further demonstrate the wider applicability of our approach to other gradient based MCMC methods, such as those based on Langevin diffusions., Conclusion: Our approach is strictly benefitial in all test cases. The Matlab sources implementing our MCMC methodology is available from https://github.com/a-kramer/ode_rmhmc.

The paper proposes Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods defined on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations. The methods provide fully automated adaptation mechanisms that circumvent the costly pilot runs that are required to tune proposal densities for Metropolis-Hastings or indeed Hamiltonian Monte Carlo and Metropolis adjusted Langevin algorithms. This allows for highly efficient sampling even in very high dimensions where different scalings may be required for the transient and stationary phases of the Markov chain. The methodology proposed exploits the Riemann geometry of the parameter space of statistical models and thus automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density. The performance of these Riemann manifold Monte Carlo methods is rigorously assessed by performing inference on logistic regression models, log-Gaussian Cox point processes, stochastic volatility models and Bayesian estimation of dynamic systems described by non-linear differential equations. Substantial improvements in the time-normalized effective sample size are reported when compared with alternative sampling approaches. MATLAB code that is available from allows replication of all the results reported. [ABSTRACT FROM AUTHOR]