MAP estimators and their consistency in Bayesian nonparametric inverse problems
By Masoumeh Dashti, Kody J. H. Law, Andrew M. Stuart, Jochen Voss. 2013
181 citations recorded by the source corpus.
We consider the inverse problem of estimating an unknown function u from noisy measurements y of a known, possibly nonlinear, map G?> applied to u. We adopt a Bayesian approach to the problem and work in a setting where the prior measure is specified as a Gaussian random field μ0. We work under a natural set of conditions on the likelihood which implies the existence of a well-posed posterior measure, μy. Under these conditions, we show that the maximum a posteriori (MAP) estimator is well defined as the minimizer of an Onsager–Machlup functional defined on the Cameron–Martin space of the prior; thus, we link a problem in probability with a problem in the calculus of variations. We then consider the case where the observational noise vanishes and establish a form of Bayesian posterior consistency for the MAP estimator. We also prove a similar result for the case where the observation of G(u)?> can be repeated as many times as desired with independent identically distributed noise. The theory is illustrated with examples from an inverse problem for the Navier–Stokes equation, motivated by problems arising in weather forecasting, and from the theory of conditioned diffusions, motivated by problems arising in molecular dynamics.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/200a651d22c180e91aa29bcabb4cd5bd5a376d29
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Artificial Intelligence
- Kolyan Ray
- Richard Nickl
- Masoumeh Dashti
- Kody J. H. Law
- Andrew M. Stuart
- Konstantinos C. Zygalakis
- Martin Burger
- Martin Benning
In Statistics and Probability
- Kolyan Ray
- Richard Nickl
- Masoumeh Dashti
- Kody J. H. Law
- Andrew M. Stuart
- Martin Burger
- Martin Benning
- Konstantinos C. Zygalakis
Citations in the corpus
- Modern regularization methods for inverse problems (2018, background): “raint) we obtain ^u 2argmin u F(Ku;f) + J(u): (12) We mention that similar reasoning in innite dimensions is not as straightforward, even the denition of the MAP estimate is a non-obvious task (cf. [136, 209]). Recent results however provide a good characterization in many relevant cases (cf. [209, ”
- Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions (2018, result): “In this regard our results are related to recent investigations of Bayesian inverse problems [29, 39, 4, 17, 28, 35], Bernstein-von Mises theorems [40, 32, 31, 33] and diffusion models [50, 38, 54, 34, 49, 26, 1].”
- The Bayesian Approach to Inverse Problems (2013, methodology): “For example, on the analysis side, the idea of MAP estimators, which links the Bayesian approach with classical regularization, developed for Gaussian priors in [30], has recently been extended to other prior models in [47]; the study of contraction of the posterior distribution to a Dirac measure o”
- Data Assimilation: A Mathematical Introduction (2015, background): “See [122] and the references therein for a discussion of the applied context; see [35] for a more theoretical presentation, including connections to the Onsager-Machlup functional arising in the theory of diffusion processes .”