Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Analysis of the Gibbs sampler for hierarchical inverse problems   Many inverse problems arising in applications come from continuum models
where the unknown parameter is a field. In practice the unknown field is
discretized resulting in a problem in $\mathbb{R}^N$, with an understanding
that refining the discretization, that is increasing $N$, will often be
desirable. In the context of Bayesian inversion this situation suggests the
importance of two issues: (i) defining hyper-parameters in such a way that they
are interpretable in the continuum limit $N \to \infty$ and so that their
values may be compared between different discretization levels; (ii)
understanding the efficiency of algorithms for probing the posterior
distribution, as a function of large $N.$ Here we address these two issues in
the context of linear inverse problems subject to additive Gaussian noise
within a hierarchical modelling framework based on a Gaussian prior for the
unknown field and an inverse-gamma prior for a hyper-parameter, namely the
amplitude of the prior variance. The structure of the model is such that the
Gibbs sampler can be easily implemented for probing the posterior distribution.
Subscribing to the dogma that one should think infinite-dimensionally before
implementing in finite dimensions, we present function space intuition and
provide rigorous theory showing that as $N$ increases, the component of the
Gibbs sampler for sampling the amplitude of the prior variance becomes
increasingly slower. We discuss a reparametrization of the prior variance that
is robust with respect to the increase in dimension; we give numerical
experiments which exhibit that our reparametrization prevents the slowing down.
Our intuition on the behaviour of the prior hyper-parameter, with and without
reparametrization, is sufficiently general to include a broad class of
nonlinear inverse problems as well as other families of hyper-priors.

23
The categories of this paper are: solv-int
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
6
The categories of this paper are: astro-ph
29
The categories of this paper are: funct-an
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
21
The categories of this paper are: adap-org
31
The categories of this paper are: ao-sci
14
The categories of this paper are: math-ph
11
The categories of this paper are: hep-lat
7
The categories of this paper are: hep-th
13
The categories of this paper are: quant-ph
19
The categories of this paper are: eess