mutual information [*per degree of freedom*]{} that is the figure of merit. We justify this by noting that the amount of mutual information obtained for some of our model observations is large, sometimes exceeding 20 bits. We argue that such a value is reasonable because it is calculated for a three-parameter atmosphere model and it thus distributed over 3 degrees of freedom, so that 20 bits of total mutual information corresponds to 6.67 bits per degree of freedom for our model over and above the prior. This corresponds to error bars 100 times narrower than the width of the prior, which is reasonable for the very broad priors we use in this study. Most measurements will yield less mutual information than this, and there may be other systematic factors that prevent reaching this precision. For a more complex forward model with more parameters, the mutual information per degree of freedom is likely to be lower, and measuring the parameters to a precision of a few percent to a few tens of percent would be plausible based on these results.

However, the difficulty with the use of mutual information is that it is computationally intensive, especially for the dense data sets 