Inversion techniques are the most powerful methods to obtain information
about the thermodynamical and magnetic properties of solar and stellar
atmospheres. In the last years, we have witnessed the development of highly
sophisticated inversion codes that are now widely applied to
spectro-polarimetric observations. The majority of these inversion codes are
based on the optimization of a complicated non-linear merit function. The large
experience gained over the years have facilitated the recovery of the model
that best fits a given observation. However, and except for the recently
developed inversion codes based on database search algorithms together with the
application of Principal Component Analysis, no reliable and statistically
well-defined confidence intervals can be obtained for the parameters inferred
from the inversions. A correct estimation of the confidence intervals for all
the parameters that describe the model is mandatory. Additionally, it is
fundamental to apply efficient techniques to assess the ability of models to
reproduce the observations and to what extent the models have to be refined or
can be simplified. Bayesian techniques are applied to analyze the performance
of the model to fit a given observed Stokes vector. The posterior distribution,
that takes into account both the information about the priors and the
likelihood, is efficiently sampled using a Markov Chain Monte Carlo method.
For simplicity, we focus on the Milne-Eddington approximate solution of the
radiative transfer equation and we only take into account the generation of
polarization through the Zeeman effect. However, the method is extremely
general and other more complex forward models can be applied, even allowing for
the presence of atomic polarization. We illustrate the ability of the method
with the aid of different problems, from academic to more realistic examples.
We show that the information provided by the posterior distribution turns out
to be fundamental to understand and determine the amount of information
available in the Stokes profiles in these particular cases.