Sampling the posterior: An approach to non-Gaussian data assimilation

By Amit Apte, Martin Hairer, Andrew M. Stuart, Jochen Voss. 2007

139 citations recorded by the source corpus.

The viewpoint taken in this paper is that data assimilation is fundamentally a statistical problem and that this problem should be cast in a Bayesian framework. In the absence of model error, the correct solution to the data assimilation problem is to find the posterior distribution implied by this Bayesian setting. Methods for dealing with data assimilation should then be judged by their ability to probe this distribution. In this paper we propose a range of techniques for probing the posterior distribution, based around the Langevin equation; and we compare these new techniques with existing methods. When the underlying dynamics is deterministic, the posterior distribution is on the space of initial conditions leading to a sampling problem over this space. When the underlying dynamics is stochastic the posterior distribution is on the space of continuous time paths. By writing down a density, and conditioning on observations, it is possible to define a range of Markov Chain Monte Carlo (MCMC) methods which sample from the desired posterior distribution, and thereby solve the data assimilation problem. The basic building-blocks for the MCMC methods that we concentrate on in this paper are Langevin equations which are ergodic and whose invariant measures give the desired distribution; in the case of path space sampling these are stochastic partial differential equations (SPDEs). Two examples are given to show how data assimilation can be formulated in a Bayesian fashion. The first is weather prediction, and the second is Lagrangian data assimilation for oceanic velocity fields. Furthermore the relationship between the Bayesian approach outlined here and the commonly used Kalman filter based techniques, prevalent in practice, is discussed. Two simple pedagogical examples are studied to illustrate the application of Bayesian sampling to data assimilation concretely. Finally a range of open mathematical and computational issues, arising from the Bayesian approach, are outlined.

Subfields: Artificial Intelligence, Atmospheric Science, Ocean Engineering

View Paper: https://www.semanticscholar.org/paper/43c222041db38a0795ccdc1dff7816ac5f71b223

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Citations in the corpus

Variational Inference for Diffusion Processes (2007, methodology): “In statistics the problem of Bayesian inference for both the state and parameters, within partially observed, non-linear diffusion processes has been tackled using Markov Chain Monte Carlo (MCMC) approaches based on data augmentation [17, 11], Monte Carlo exact simulation methods [6], or Langevin / ”
Parameter estimation for differential equations: a generalized smoothing approach (2007, unknown): “This approach has been used for the problem of data assimilation in Apte et al. (2007), where they use criteria that are closely related to our own equation (14).”
Data assimilation: Mathematical and statistical perspectives (2008, methodology): “We study time-dependent problems in which the desired unknown is either the initial condition (a function of space alone) or a time-dependent function (a function of both space and time) [5].”
Gaussian Process Approximations of Stochastic Differential Equations (2007, methodology): “Future work will explore further the links between our methods and the cutting edge data assimilation methods developped by Eyink et al. (2004); Apte et al. (2006) and will focus on a better smoothing algorithm.”
Inverse problems: A Bayesian perspective (2010, methodology): “In Apte et al. (2007) and Apte, Jones and Stuart (2008a), studies of the ideal solution obtained from applying MCMC methods to the posterior (2.24) are compared with ensemble Kalman filter methods.”
Bayesian inverse problems for functions and applications to fluid mechanics (2009, methodology): “In [1, 2] we include preliminary studies comparing the ideal solution with ensemble Kalman filter methods applied to Lagrangian data driven by a low-dimensional truncation of the linearized shallow water equations.”
Bayesian inference with optimal maps (2011, methodology): “A non-exhaustive list of recent methodological advances includes adaptivity based on past samples [11, 12], Langevin methods [13, 14], the use of Hessian information [15] and Hessians with low-rank structure [16], Hamiltonian dynamics [17], the exchange of information among multiple chains [18– 20],”
ANALYSIS OF SPDES ARISING IN PATH SAMPLING PART II: THE NONLINEAR CASE (2006, unknown): “5/6, 1657–1706 DOI: 10.1214/07-AAP441 c© Institute of Mathematical Statistics, 2007”