Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

Tronarp, Filip; Kersting, Hans; Särkkä, Simo; Hennig, Philipp

doi:10.1007/s11222-019-09900-1

Cited by 44 publications

(84 citation statements)

References 54 publications

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“…In this paper, we determine the convergence rates of a recent family of PN methods (Schober et al 2014;Kersting and Hennig 2016;Magnani et al 2017;Schober et al 2019;Tronarp et al 2019) which recast an IVP as a stochastic filtering problem (Øksendal 2003, Chapter 6), an approach that has been studied in other settings (Jazwinski 1970), but has not been applied to IVPs before. These methods assume a priori that the solution x and its first q ∈ N derivatives follow a Gauss-Markov process X that solves a stochastic differential equation (SDE).…”

Section: Introductionmentioning

confidence: 99%

Convergence rates of Gaussian ODE filters

2020

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A recently introduced class of probabilistic (uncertainty-aware) solvers for ordinary differential equations (ODEs) applies Gaussian (Kalman) filtering to initial value problems. These methods model the true solution x and its first q derivatives a priori as a Gauss–Markov process $${\varvec{X}}$$ X , which is then iteratively conditioned on information about $${\dot{x}}$$ x ˙ . This article establishes worst-case local convergence rates of order $$q+1$$ q + 1 for a wide range of versions of this Gaussian ODE filter, as well as global convergence rates of order q in the case of $$q=1$$ q = 1 and an integrated Brownian motion prior, and analyses how inaccurate information on $${\dot{x}}$$ x ˙ coming from approximate evaluations of f affects these rates. Moreover, we show that, in the globally convergent case, the posterior credible intervals are well calibrated in the sense that they globally contract at the same rate as the truncation error. We illustrate these theoretical results by numerical experiments which might indicate their generalizability to $$q \in \{2,3,\ldots \}$$ q ∈ { 2 , 3 , … } .

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Section: Introductionmentioning

confidence: 99%

Convergence rates of Gaussian ODE filters

2020

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“…In such a problem, a discrete-time state space model is obtained by discretization of continuous differential equations, and the transition model p(x t+1 |x t ), which is probabilistic, characterizes numerical uncertainties caused by discretization errors. Importantly, certain numerical solvers of differential equations based on probabilistic numerical methods (Hennig et al, 2015;Cockayne et al, 2019;Oates and Sullivan, 2019) provide the transition model p(x t+1 |x t ) in terms of Gaussian probabilities (Schober et al, 2014;Kersting and Hennig, 2016;Schober et al, 2018;Tronarp et al, 2018). Hence, we expect that it is possible to use a transition model obtained from such probabilistic solvers with the Mb-KSR, and to combine a time-series model described by differential equations with nonparametric kernel Bayesian inference.…”

Section: Discussionmentioning

confidence: 99%

Model-based kernel sum rule: kernel Bayesian inference with probabilistic models

et al. 2020

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Kernel Bayesian inference is a principled approach to nonparametric inference in probabilistic graphical models, where probabilistic relationships between variables are learned from data in a nonparametric manner. Various algorithms of kernel Bayesian inference have been developed by combining kernelized basic probabilistic operations such as the kernel sum rule and kernel Bayes' rule. However, the current framework is fully nonparametric, and it does not allow a user to flexibly combine nonparametric and model-based inferences. This is inefficient when there are good probabilistic models (or simulation models) available for some parts of a graphical model; this is in particular true in scientific fields where "models" are the central topic of study. Our contribution in this paper is to introduce a novel approach, termed the model-based kernel sum rule (Mb-KSR), to combine a probabilistic model and kernel Bayesian inference. By combining the Mb-KSR with the existing kernelized probabilistic rules, one can develop various algorithms for hybrid (i.e., nonparametric and model-based) inferences. As an illustrative example, we consider Bayesian filtering in a state space model, where typically there exists an accurate probabilistic model for the state transition process. We propose a novel filtering method that combines model-based inference for

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“…1.2.6 , Tronarp et al [2019] The survey just presented begs the question of whether a Bayesian PNM for ODEs can exist at all. A first step toward this goal was taken in , where it was argued that an information operator can be constructed if the vector field f is brought to the left-hand-side in Eq.…”

Section: Chkrebtii Et Al [2016]mentioning

confidence: 99%

“…Indeed, unless f depends linearly on its second argument and conjugacy properties of the prior can be exploited, the posterior cannot easily be characterised. Approximate techniques from nonlinear filtering were proposed to address this challenge in Tronarp et al [2019].…”

Section: Chkrebtii Et Al [2016]mentioning

confidence: 99%

A Role for Symmetry in the Bayesian Solution of Differential Equations

Wang¹,

Cockayne²,

Oates³

2020

Bayesian Anal.

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The interpretation of numerical methods, such as finite difference methods for differential equations, as point estimators suggests that formal uncertainty quantification can also be performed in this context. Competing statistical paradigms can be considered and Bayesian probabilistic numerical methods (PNMs) are obtained when Bayesian statistical principles are deployed. Bayesian PNM have the appealing property of being closed under composition, such that uncertainty due to different sources of discretisation in a numerical method can be jointly modelled and rigorously propagated. Despite recent attention, no exact Bayesian PNM for the numerical solution of ordinary differential equations (ODEs) has been proposed. This raises the fundamental question of whether exact Bayesian methods for (in general nonlinear) ODEs even exist. The purpose of this paper is to provide a positive answer for a limited class of ODE. To this end, we work at a foundational level, where a novel Bayesian PNM is proposed as a proof-of-concept. Our proposal is a synthesis of classical Lie group methods, to exploit underlying symmetries in the gradient field, and non-parametric regression in a transformed solution space for the ODE. The procedure is presented in detail for first and second order ODEs and relies on a certain strong technical condition -existence of a solvable Lie algebra -being satisfied. Numerical illustrations are provided.

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Probabilistic solutions to ordinary differential equations as nonlinear Bayesian filtering: a new perspective

Cited by 44 publications

References 54 publications

Convergence rates of Gaussian ODE filters

Convergence rates of Gaussian ODE filters

Model-based kernel sum rule: kernel Bayesian inference with probabilistic models

A Role for Symmetry in the Bayesian Solution of Differential Equations

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