BAYESIAN INVERSE PROBLEMS WITH GAUSSIAN PRIORS

By B. Knapik, A. van der Vaart, J. H. Zanten. 2011

248 citations recorded by the source corpus.

The posterior distribution in a nonparametric inverse problem is shown to contract to the true parameter at a rate that depends on the smoothness of the parameter, and the smoothness and scale of the prior. Correct combinations of these characteristics lead to the minimax rate. The frequentist coverage of credible sets is shown to depend on the combination of prior and true parameter, with smoother priors leading to zero coverage and rougher priors to conservative coverage. In the latter case credible sets are of the correct order of magnitude. The results are numerically illustrated by the problem of recovering a function from observation of a noisy version of its primitive.

Subfields: Artificial Intelligence, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/b28f3da3f56729582221cfd0caa461e55c794be9

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Citations in the corpus

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].”
MAP estimators and their consistency in Bayesian nonparametric inverse problems (2013, background): “n Bayesian nonparametric statistics has a long history [15]. The study of posterior consistency for the Bayesian approach to inverse problems is starting to receive considerable attention. The papers [23, 1] are devoted to obtaining rates of convergence for linear inverse problems with conjugate Gau”
Adaptive Bernstein–von Mises theorems in Gaussian white noise (2014, background): “This dramatically oversmoothes the truth resulting in far too narrow credible bands and is highly dangerous since it is known that oversmoothing the truth can yield zero coverage [27, 29].”
Bernstein - von Mises Theorem for growing parameter dimension (2013, background): “Some examples can be found in Cox (1993), Freedman (1999), Bontemps (2011), Knapik et al. (2011) among others.”
Nonparametric Bernstein–von Mises theorems in Gaussian white noise (2012, background): “In contrast to C′′ n , however, the credible sets in [25] are conservative in the sense”
Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator (2015, background): “Type (T2) and (T3) priors have been studied with fixed hyper-parameters by [5, 7, 15, 31, 34] or using a prior on λ = τ and λ = α in [4, 14, 18, 29].”
Statistical guarantees for Bayesian uncertainty quantification in nonlinear inverse problems with Gaussian process priors (2020, background): “Regarding statistical (frequentist) properties of posterior measures, the case of linear G is again fairly well understood due to the explicit Gaussian structure of the posterior distribution, we refer here only to [35, 52, 2, 34, 39, 22, 26] and references therein.”
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.”
A Bernstein–von Mises theorem for smooth functionals in semiparametric models (2013, background): “[26], Castillo [11] and Kruijer and Rousseau [27].”
Parallel and Flexible Sampling from Autoregressive Models via Langevin Dynamics (2021, background): “Bayesian inverse problems are explored extensively in theoretical settings, where the prior is given by a simple analytical distribution (Tropp & Wright, 2010; Knapik et al., 2011; Wang et al., 2017).”
Bayes procedures for adaptive inference in inverse problems for the white noise model (2012, methodology): “Themore recent papers [22] and [1] study the inverse problemcase, but also obtain non-adaptive results only.”
Supremum Norm Posterior Contraction and Credible Sets for Nonparametric Multivariate Regression (2014, background): “Following [18], we consider the true function f0(x) = √ 2 ∑∞ i=1 i−3/2 sin i cos{(i − 1/2)πx}, x ∈ [0,1], which has smoothness α = 1.”
Bayesian Nonparametric Causal Inference: Information Rates and Learning Algorithms (2017, unknown): “, possess frequentist coverage [7], [36], [52].”
Bayesian Recovery of the Initial Condition for the Heat Equation (2011, methodology): “In contrast to Knapik et al. (2011) however, the regularity of the prior does not have to match exactly the regularity of the truth.”