Bayesian Recovery of the Initial Condition for the Heat Equation
By B. Knapik, Van der Vaart, V. Zanten. 2011
66 citations recorded by the source corpus.
We study a Bayesian approach to recovering the initial condition for the heat equation from noisy observations of the solution at a later time. We consider a class of prior distributions indexed by a parameter quantifying “smoothness” and show that the corresponding posterior distributions contract around the true parameter at a rate that depends on the smoothness of the true initial condition and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the optimal minimax rate. One type of priors leads to a rate-adaptive Bayesian procedure. The frequentist coverage of credible sets is shown to depend on the combination of the prior and true parameter as well, with smoother priors leading to zero coverage and rougher priors to (extremely) conservative results. In the latter case, credible sets are much larger than frequentist confidence sets, in that the ratio of diameters diverges to infinity. The results are numerically illustrated by a simulated data example.
Subfields: Mathematical Physics, Statistical and Nonlinear Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/d3acb340f205fea0c9b85170f5c596809a5fab34
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Mathematical Physics
- Andrew M. Stuart
- Botond Szab'o
- Masoumeh Dashti
- Kolyan Ray
- J. H. Zanten
- Subhashis Ghosal
- A. van der Vaart
- Omiros Papaspiliopoulos
In Statistical and Nonlinear Physics
In Statistics and Probability
- Kolyan Ray
- Botond Szab'o
- Richard Nickl
- A. van der Vaart
- Judith Rousseau
- Andrew M. Stuart
- Subhashis Ghosal
- Masoumeh Dashti
Citations in the corpus
- Bayesian inverse problems with non-conjugate priors (2012, methodology): “[11, 13, 16, 29, 31] for some examples), but the situation of inverse problems has only recently been considered and then only in the conjugate setting [1, 20, 21], where explicit posterior expressions are available.”
- Analysis of the Gibbs Sampler for Hierarchical Inverse Problems (2013, background): “We consider the setting of [26, 4], that is, a similar situation with the previous example, where instead of having ( µ Kj ) 2 j − 4 we now have ( µ Kj ) 2 e − 2 sj b , for b, s > 0.”
- Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, background): “The few papers in this area include [1,14,22,23,30].”
- The Bayesian Approach to Inverse Problems (2013, background): “Recovering the truth underlying the data from the Bayesian approach, known as Bayesian posterior consistency, is the topic of [3,55]; generalizations to severely ill-posed problems, such as the heat equation, may be found in [4, 56].”
- Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “[18, 19] showed that for sequences with specific smoothness, by deliberately undersmoothing the prior, coverage of credible sets may be guaranteed.”