20 results found
Teymur O, Calderhead B, Lie HC, et al., Implicit probabilistic integrators for ODEs, Neural Information Processing Systems, Publisher: Massachusetts Institute of Technology Press, ISSN: 1049-5258
We introduce a family of implicit probabilistic integrators for initial value problems(IVPs), taking as a starting point the multistep Adams–Moulton method. Theimplicit construction allows for dynamic feedback from the forthcoming time-step, in contrast to previous probabilistic integrators, all of which are based onexplicit methods. We begin with a concise survey of the rapidly-expanding field ofprobabilistic ODE solvers. We then introduce our method, which builds on andadapts the work of Conrad et al. (2016) and Teymur et al. (2016), and provide arigorous proof of its well-definedness and convergence. We discuss the problem ofthe calibration of such integrators and suggest one approach. We give an illustrativeexample highlighting the effect of the use of probabilistic integrators—includingour new method—in the setting of parameter inference within an inverse problem.
Chkrebtii OA, Campbell DA, Calderhead B, et al., 2016, Bayesian Solution Uncertainty Quantification for Differential Equations, Bayesian Analysis, Vol: 11, Pages: 1239-1267, ISSN: 1936-0975
We explore probability modelling of discretization uncertainty for systemstates defined implicitly by ordinary or partial differential equations. Accountingfor this uncertainty can avoid posterior under-coverage when likelihoods areconstructed from a coarsely discretized approximation to system equations. A formalismis proposed for inferring a fixed but a priori unknown model trajectorythrough Bayesian updating of a prior process conditional on model information.A one-step-ahead sampling scheme for interrogating the model is described, itsconsistency and first order convergence properties are proved, and its computationalcomplexity is shown to be proportional to that of numerical explicit one-stepsolvers. Examples illustrate the flexibility of this framework to deal with a widevariety of complex and large-scale systems. Within the calibration problem, discretizationuncertainty defines a layer in the Bayesian hierarchy, and a Markovchain Monte Carlo algorithm that targets this posterior distribution is presented.This formalism is used for inference on the JAK-STAT delay differential equationmodel of protein dynamics from indirectly observed measurements. The discussionoutlines implications for the new field of probabilistic numerics.
Epstein M, Calderhead B, Girolami MA, et al., 2016, Bayesian Statistical Inference in Ion-Channel Models with Exact Missed Event Correction, BIOPHYSICAL JOURNAL, Vol: 111, Pages: 333-348, ISSN: 0006-3495
Groen D, Calderhead B, 2015, Science hackathons for developing interdisciplinary research and collaborations, eLife, Vol: 4, ISSN: 2050-084X
Science hackathons can help academics, particularly those in the early stage of their careers, to build collaborations and write research proposals.
Knapp B, Bardenet R, Bernabeu MO, et al., 2015, Ten Simple Rules for a Successful Cross-Disciplinary Collaboration, PLOS Computational Biology, Vol: 11, ISSN: 1553-734X
Calderhead B, 2014, A general construction for parallelizing Metropolis-Hastings algorithms, PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, Vol: 111, Pages: 17408-17413, ISSN: 0027-8424
Wang Y, Huang K, Cao Z, et al., 2014, Bayesian identification of soil strata in London Clay, GEOTECHNIQUE, Vol: 64, Pages: 1014-1016, ISSN: 0016-8505
Smith MJ, Palmer PI, Purves DW, et al., 2014, CHANGING HOW EARTH SYSTEM MODELING IS DONE TO PROVIDE MORE USEFUL INFORMATION FOR DECISION MAKING, SCIENCE, AND SOCIETY, BULLETIN OF THE AMERICAN METEOROLOGICAL SOCIETY, Vol: 95, Pages: 1453-1464, ISSN: 0003-0007
Kramer A, Calderhead B, Radde N, 2014, Hamiltonian Monte Carlo methods for efficient parameter estimation in steady state dynamical systems, BMC BIOINFORMATICS, Vol: 15, ISSN: 1471-2105
BackgroundParameter 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.ResultsWe 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
Osborne JM, Bernabeu MO, Bruna M, et al., 2014, Ten Simple Rules for Effective Computational Research, PLOS COMPUTATIONAL BIOLOGY, Vol: 10
Calderhead B, Sustik M, 2013, Sparse Approximate Manifolds for Differential Geometric MCMC, 26th Annual Conference on Neural Information Processing Systems 2012, Publisher: Curran Associates Inc., Pages: 2879-2887
Calderhead B, Epstein M, Sivilotti L, et al., 2013, Bayesian approaches for mechanistic ion channel modeling., Methods Mol Biol, Vol: 1021, Pages: 247-272
We consider the Bayesian analysis of mechanistic models describing the dynamic behavior of ligand-gated ion channels. The opening of the transmembrane pore in an ion channel is brought about by conformational changes in the protein, which results in a flow of ions through the pore. Remarkably, given the diameter of the pore, the flow of ions from a small number of channels or indeed from a single ion channel molecule can be recorded experimentally. This produces a large time-series of high-resolution experimental data, which can be used to investigate the gating process of these channels. We give a brief overview of the achievements and limitations of alternative maximum-likelihood approaches to this type of modeling, before investigating the statistical issues associated with analyzing stochastic model reaction mechanisms from a Bayesian perspective. Finally, we compare a number of Markov chain Monte Carlo algorithms that may be used to tackle this challenging inference problem.
Mohamed L, Calderhead B, Filippone M, et al., 2012, Population MCMC methods for history matching and uncertainty quantification, 12th European Conference on the Mathematics of Oil Recovery (ECMOR), Publisher: SPRINGER, Pages: 423-436, ISSN: 1420-0597
Calderhead B, Girolami M, 2011, Statistical analysis of nonlinear dynamical systems using differential geometric sampling methods, INTERFACE FOCUS, Vol: 1, Pages: 821-835, ISSN: 2042-8898
Girolami M, Calderhead B, 2011, Riemann manifold Langevin and Hamiltonian Monte Carlo methods, JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B-STATISTICAL METHODOLOGY, Vol: 73, Pages: 123-214, ISSN: 1369-7412
Girolami M, Calderhead B, Vyshemirsky V, 2009, System Identification and Model Ranking: The Bayesian Perspective Learning and Inference, Learning and Inference in Computational Systems Biology, Editors: Lawrence, Girolami, Rattray, Sanguinetti, ISBN: 9780262013864
Calderhead B, Girolami M, 2009, Estimating Bayes factors via thermodynamic integration and population MCMC, COMPUTATIONAL STATISTICS & DATA ANALYSIS, Vol: 53, Pages: 4028-4045, ISSN: 0167-9473
Calderhead B, Girolami M, Lawrence N, 2009, Accelerating Bayesian inference over nonlinear differential equations with Gaussian processes
Luo X, Calderhead B, Liu H, et al., 2007, On the initial configurations of collapsible channel flow, 4th MIT Conference on Computational Fluid and Solid Mechanics, Publisher: PERGAMON-ELSEVIER SCIENCE LTD, Pages: 977-987, ISSN: 0045-7949
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