Non-Gaussian statistical inverse problems. Part I: Posterior distributions
By Sari Lasanen. 2012
57 citations recorded by the source corpus.
One approach to noisy inverse problems is to use Bayesian methods. In this work, the statistical inverse problem of estimating the probability distribution of an infinite-dimensional unknown given its noisy indirect infinite-dimensional observation is studied in the Bayesian framework. The motivation for the work arises from the fact that the Bayesian computations are usually carried out in finite-dimensional cases, while the original inverse problem is often infinite-dimensional. A good understanding of an infinite-dimensional problem is, in general, helpful in finding efficient computational approaches to the problem. The fundamental question of well-posedness of the infinite-dimensional statistical inverse problem is considered. In particular, it is shown that the continuous dependence of the posterior probabilities on the realizations of the observation provides a certain degree of uniqueness for the posterior distribution. Special emphasis is on finding tools for working with non-Gaussian noise models. Especially, the applicability of the generalized Bayes formula is studied. Several examples of explicit posterior distributions are provided.
Subfields: Artificial Intelligence, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/ae95e567841107ce208791c89bce2413244af381
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In Artificial Intelligence
- Masoumeh Dashti
- Sari Lasanen
- Andrew M. Stuart
- Lassi Roininen
- Janne M. J. Huttunen
- Mark A. Girolami
- G. Em Karniadakis
- Liang Yan
In Statistics and Probability
- Masoumeh Dashti
- Sari Lasanen
- Andrew M. Stuart
- Lassi Roininen
- Janne M. J. Huttunen
- Mark A. Girolami
- Richard Nickl
- Chris J. Oates
Citations in the corpus
- Whittle-Matérn priors for Bayesian statistical inversion with applications in electrical impedance tomography (2014, background): “1For and definition of posterior distributions and the required conditions for Bayes’ formula in infinite-dimensional spaces, see Lasanen [13].”
- The Bayesian Approach to Inverse Problems (2013, background): “ayes’ Theorem for inverse problems on function space, along the lines described here, may be found in [18, 93]. The reader is also directed to the papers [62, 63] for earlier related material, and to [64, 65, 66] for recent developments. • Subsection 3.3. The inverse problem for the heat equation wa”