in the forward model. If the lengths of [**x**]{} and [**y**]{} are the same then (1) may be readily inverted to $${\bf x}={\bf x_{a}}+{\bf K^{-1}({\bf y} -F({\bf x_a}))}$$ Real data are often noisy and usually have a large number of measurements that over constrain the atmospheric state. For this we must use a more sophisticated scheme to invert the data to determine the atmospheric properties. This can be readily achieved by using a Bayesian framework. In the remainder of this section, we present the basic formalism and useful equations and algorithms that we can use to retrieve atmospheric properties from spectra as well as their information content, following the derivations in Rodgers (2000). For further details, see either Rodgers (2000) or Jacob (2007).

Bayes theorem can be written as $$P({\bf x}|{\bf y}) \propto P({\bf y}|{\bf x})P({\bf x})$$ where P([**x**]{}) is the prior probability distribution, which is knowledge of the atmospheric state before making a measurement, P([**y**]{}$|$[**x**]{}) is the likelihood function, that is the probability that the data exists within the context of a particular model, and P([**x**]{}$|$[**y**]{}) is the posterior probability distribution density function which can be interpreted as the probability that some state [**x**]{}, in our case atmospheric state, 