et al. 2008; Madhusudhan & Seager 2009)

Others have used multi-dimensional grid models to constrain atmospheric parameters (Madhusudhan & Seager 2009), a method that is well tuned to systematically searching the parameter space given sparse data (as with Spitzer IRAC color photometry). In this approach, an ensemble of forward models are generated using up to 10 gridded free parameters (6 to govern the shape of the temperature profile and 4 scaling factors for uniform mixing ratios of H$_2$O, CH$_4$, CO, and CO$_2$); model families that best describe the data are selected based on a chi-squared statistic criterion. Because of the degeneracies between the different gases, and between gases and temperature, thousands of solutions can exist within a given chi-squared region, thus only giving loose constraints on the atmospheric composition and temperature. Furthermore, the formalism provides no easy way to explore the change in information content associated with a change in the data phase space (e.g., $R$ or $S/N$).

Here, we present the [*inverse*]{} approach (see also Lee et al. 2011) that determines the atmospheric “state” (i.e. its temperature structure and abundances) by minimizing a cost function that simultaneously takes into account new measurements and prior knowledge of atmospheric properties (such 