# Scale Invariant Markov Models for Bayesian Resolution of Inverse
Problems

## Stéphane Brette, Jérôme Idier and Ali
Mohammad-Djafari

Laboratoire des Signaux et Systèmes (CNRS-ESE-UPS)

École Supérieure d'Électricité,

Plateau de Moulon, 91192 Gif-sur-Yvette Cedex, France

### Abstract

In a Bayesian estimation approach for solving inverse problems we need to
specify the prior law **p(x;\theta)** and
the conditional law **p(y|x;\theta)** for calculation of the
posterior law
**p(x|y;\theta)**. We need also a cost function
**C(\widehat{x},x)** to define
an estimator **\widehat{x}(y,\theta)** depending on the data
**y**
and the hyperparameters **\theta**.
Except in the linear Gaussian case where all the classical Bayesian estimators
become a unique linear function of the data, most Bayesian estimators
are otherwise nonlinear functions of the data, and so dependent on the
measurement scale.
When dealing with linear inverse problems linearity is sometimes
too strong a property, while *scale invariance property*
(SIP) often remains a desirable one.
In this paper, first we investigate general conditions on classes of
Bayesian estimators which satisfy the SIP and their consequences on
the cost function and prior laws.
Then we show that the cost functions of the three classical Bayesian
estimators satisfy the SIP constraints.
Finally we discuss how to choose the prior laws to obtain scale
invariant Bayesian estimators. For this, we consider two cases of prior laws:
*entropic prior laws* and *first-order Markov models*.
In related preceding works `MaxEnt91`, `MaxEnt93`,
the SIP constraints have been studied for the case of entropic prior laws.
In this paper we extend that work to the case of first-order Markov
models and show that the SIP constrains the potential functions of
the posterior laws to be in one of the following forms:

We also investigate the application of further constraints such as
symmetry, unimodality and convexity of the clique potentials
**\phi(x_s, x_r)** and show that only some of the Markov
models in use for modern imaging purposes belong to the exhibited
classes.

### Keywords

Bayesian estimation, Scale invariance, Inverse Problems,
Entropic prior laws, Markov models, Image reconstruction

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