Bayesian Solution Uncertainty Quantification for Differential Equations
Bayesian Solution Uncertainty Quantification for Differential Equations
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DOI：
10.1214/16-ba1017
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发表时间：
2016-12-01
影响因子：
4.4
通讯作者：
Girolami, Mark A.
中科院分区：
文献类型：
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作者：
Chkrebtii, Oksana A.;Campbell, David A.;Girolami, Mark A.
我们探讨了由普通或部分微分方程隐式定义的系统状态离散不确定性的概率建模。考虑到这种不确定性的核算可以避免后覆盖后覆盖后，当可能性离散的近似与系统方程式构建时。提出了一种形式主义，用于通过贝叶斯更新模型信息来推断固定但先验未知的模型轨迹。描述了一种用于询问模型的一步采样方案，证明其一致性和一阶收敛属性，并显示其计算复杂性与数值显式的一步求解器成正比。示例说明了此框架的灵活性，以处理各种复杂和大规模的系统。在校准问题中，离散化不确定性定义了贝叶斯层次结构中的一层，并且提出了针对此后验分布的马尔可夫链蒙特卡洛算法。这种形式主义用于推断从间接观察到的测量值的蛋白质动力学的JAK-STAT延迟微分方程模型。讨论概述了对概率数字新领域的影响。
We explore probability modelling of discretization uncertainty for system states defined implicitly by ordinary or partial differential equations. Accounting for this uncertainty can avoid posterior under-coverage when likelihoods are constructed from a coarsely discretized approximation to system equations. A formalism is proposed for inferring a fixed but a priori unknown model trajectory through Bayesian updating of a prior process conditional on model information. A one-step-ahead sampling scheme for interrogating the model is described, its consistency and first order convergence properties are proved, and its computational complexity is shown to be proportional to that of numerical explicit one-step solvers. Examples illustrate the flexibility of this framework to deal with a wide variety of complex and large-scale systems. Within the calibration problem, discretization uncertainty defines a layer in the Bayesian hierarchy, and a Markov chain Monte Carlo algorithm that targets this posterior distribution is presented. This formalism is used for inference on the JAK-STAT delay differential equation model of protein dynamics from indirectly observed measurements. The discussion outlines implications for the new field of probabilistic numerics.