2
$\begingroup$

I am getting started with inverse problems in statistics. However, I didn't something related to it.

I was reading this paper http://math.uni-heidelberg.de/studinfo/reiss/CavalierInvProb.pdf.

It says

The classical problem is the following : let A be an operator from the Hilbert space H in G :

Given g ∈ G find f ∈ H such that Af = g.

This is really an inverse problem in the sense that one has to invert the operator A. A case of major interest is the case of ill-posed problems where the operator is not invertible. The problem is then to handle this inversion in order to obtain a precise reconstruction.

A problem is well posed if

  1. there exists a solution to the problem (existence)

  2. there is at most one solution to the problem (uniqueness)

  3. the solution depends continuously on the data (stability)

A problem which is not well-posed is called ill-posed.

If the data space is defined as the set of solutions, existence is clear. However, this could be modified if the data are perturbed by noise. Uniqueness of the solution is not easy to show. In case where it is not garanted by the data, then the set of a priori solutions can be restricted, and the problem is then reformulated. Nevertheless, the main issue is usually stability. Indeed, suppose $A^{−1}$ exists but is not bounded. Given a noisy version of g called $g_ε$, the reconstruction $f_ε$ = $A^{−1}g_ε$ may be far from the true f.

I actually didn't get it. Can you give some examples to help me clarify. I didn't get what they mean by $A^{-1}$ exists but is not bounded. And what about the uniqueness of the solution. Can you give an example where the solution is not unique. I am not being able to grasp the concept.

$\endgroup$
2
  • 4
    $\begingroup$ Judging from this quotation, the paper is written at a very abstract level and is intended for people who are (a) knowledgeable about statistical problems in the first place and (b) conversant with the theory of Hilbert and Banach spaces and functional analysis. It sounds like this does not match your background--your questions have to do with linear operators, not statistics--and so you would be better off reading a different introduction to statistics. Might I suggest searching our site for recommendations? $\endgroup$ – whuber Mar 10 '13 at 16:30
  • 1
    $\begingroup$ @whuber. Can you suggest some? I want to learn about the whole inverse problem in statistics $\endgroup$ – user34790 Mar 10 '13 at 23:03