produced by [*JWST*]{}. To calculate it requires Monte Carlo integration over both the model parameters and the synthetic observations. Therefore, in order to simplify the calculation, we employ the approximation $$\begin{aligned}
I(\Theta,X) &\approx H(\Theta) - H(\Theta|X) \approx H(\Theta) - H(\Theta|x) \nonumber \\
            &= - \int_\Theta p(\theta)\log_2 p(\theta)d(\theta) + \int_\Theta p(\theta|x) \log_2 p(\theta|x) d\theta.\end{aligned}$$ This is just the difference in entropy between the prior and the posterior for a single simulated observation rather than the average, computed without marginalizing over the evidence. This is also the metric used by @2012ApJ...749...93L in their analysis. Since the observation contains many (hundreds or more) data points, the effect of outliers on the conditional entropy estimate should mostly damp out. We estimate the accuracy of this approximation in Section \[error\] by computing a low-resolution Monte Carlo integral of the full mutual information for comparison. We do this by computing many posterior distributions from different random seeds with a Gaussian distribution and averaging the results. Since the formula using a single posterior distribution is an approximation to the correct value, we use it as an estimate, which we define as the [*posterior entropy method*]{}.

We also have a second method to estimate the mutual information. For 