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Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems.

Kramer A, Calderhead B, Radde N - BMC Bioinformatics (2014)

Bottom Line: 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.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.Our approach is strictly benefitial in all test cases.

View Article: PubMed Central - PubMed

Affiliation: Institute for Systems Theory and Automatic Control, Pfaffenwaldring 9, 70550 Stuttgart, Germany. andrei.kramer@ist.uni-stuttgart.de.

ABSTRACT

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.

Show MeSH
MAPK posterior comparisons fltr. (1) posterior inferred from RMHMC sample. (2) Newton Raphson driven RMHMC posterior inferred from sample. (3) posterior inferred from NR-HMC sample (4) exact posterior on a grid. We used a kernel density estimator (kde) to infer densities from samples.
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Fig1: MAPK posterior comparisons fltr. (1) posterior inferred from RMHMC sample. (2) Newton Raphson driven RMHMC posterior inferred from sample. (3) posterior inferred from NR-HMC sample (4) exact posterior on a grid. We used a kernel density estimator (kde) to infer densities from samples.

Mentions: We used a normal prior in the logarithmic parameter space which covers several orders of magnitude for both parameters. This example has two parameters, making it convenient to compare the kernel density estimate of the posteriors to an evaluation of the posterior on a regular grid, which removes any sampling related errors. This comparison can be seen in FigureĀ 1. All posteriors look very similar, which indicates proper convergence for all three algorithms.Figure 1


Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems.

Kramer A, Calderhead B, Radde N - BMC Bioinformatics (2014)

MAPK posterior comparisons fltr. (1) posterior inferred from RMHMC sample. (2) Newton Raphson driven RMHMC posterior inferred from sample. (3) posterior inferred from NR-HMC sample (4) exact posterior on a grid. We used a kernel density estimator (kde) to infer densities from samples.
© Copyright Policy - open-access
Related In: Results  -  Collection

License 1 - License 2
Show All Figures
getmorefigures.php?uid=PMC4262080&req=5

Fig1: MAPK posterior comparisons fltr. (1) posterior inferred from RMHMC sample. (2) Newton Raphson driven RMHMC posterior inferred from sample. (3) posterior inferred from NR-HMC sample (4) exact posterior on a grid. We used a kernel density estimator (kde) to infer densities from samples.
Mentions: We used a normal prior in the logarithmic parameter space which covers several orders of magnitude for both parameters. This example has two parameters, making it convenient to compare the kernel density estimate of the posteriors to an evaluation of the posterior on a regular grid, which removes any sampling related errors. This comparison can be seen in FigureĀ 1. All posteriors look very similar, which indicates proper convergence for all three algorithms.Figure 1

Bottom Line: 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.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.Our approach is strictly benefitial in all test cases.

View Article: PubMed Central - PubMed

Affiliation: Institute for Systems Theory and Automatic Control, Pfaffenwaldring 9, 70550 Stuttgart, Germany. andrei.kramer@ist.uni-stuttgart.de.

ABSTRACT

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.

Show MeSH