# The maximum entropy on the mean method, noise and sensitivity

## J.-F. Bercher, G. Le Besnerais and G. Demoment

Laboratoire des signaux et systèmes,

Plateau de Moulon 91190 Gif-sur-Yvette, France

### Abstract

In this paper, we discuss some developments in the Maximum Entropy on
the Mean Method (`MEMM`), which was introduced by J. Navaza
(1985), and further studied by D. Dacunha-Castelle and F. Gamboa
(1989). They concern accounting for noisy data and the derivation of
error bars on the reconstruction.
The `MEMM` enables one to solve the linear ill-posed problem
**y=Ax** when prior knowledge or constraints are added in
the form **x \in {\cal C}**, where **{\cal
C}** is a convex set endowed with a measure
**\nu**. It may be shown, using tools of convex analysis,
that the solution **\hat{x}** can be obtained as
**\hat{x}= \argmin{x}{{\cal F}(x)}** submitted to
**y=Ax**, where **{\cal F}** is a convex
functional built from the knowledge of **{\cal C}** and
**\nu**. With different reference measures
**\nu**, classical criteria are obtained, such as
least-squares, Shannon entropy, Burg entropy, Fermi-Dirac
entropy... In some cases, however, **{\cal F}** has no
explicit form and the problem has to be solved from its dual
formulation. These basics of the `MEMM` are summarized in
the first part of the paper; further details may be found in the
references therein.

The presence of additive noise on the data is considered in a second
part. After having reviewed some common approaches, we show that
additive noise may be properly integrated in the `MEMM`
scheme. The object and noise are searched for simultaneously in a new
convex **{\cal C}'**, which includes the previous convex
**{\cal C}**. The reference measure of this new set is
built from the prior knowledge on the solution and from the noise
distribution. The `MEMM` solution is now obtained as
**\hat{x}=\argmin{x}{\left({\cal F}(x) + {\cal
G}(y-Ax)\right)}**, where **{\cal F}**
and **{\cal G}** are convex functionals. The functional
**{\cal G}** is related to the prior distribution of
noise, and may be used to take into account specific non gaussian
noise distributions.

Thanks to the regularity of the criteria, the sensitivity of the solution to
variations of the data may be obtained from second order derivatives of the
criterion at the stationary point.

MaxEnt 94 Abstracts / mas@mrao.cam.ac.uk