$$H(X) = -\int_{x\in X}p(x)\log_2 p(x)dx.$$

One bit of information is a significant gain in a field where few definite conclusions can be drawn with confidence. For a Gaussian distribution, the entropy is $H(X) = \log_2(\sigma_X\sqrt{2\pi e})$, so a difference of one bit in entropy approximately corresponds to a factor of 2 in $\sigma$. For example, one of the prior distributions we employ in our study specifies an atmosphere metallicity of $0.5 \pm 0.5$ dex, or more specifically, a normal distribution with $\mu_{\log Z} = 0.5$ dex and $\sigma_{\log Z} = 0.5$ dex. If an observing program provides one bit of information, then we could use it to measure the metallicity of a planet’s atmosphere to a precision of $\sigma_{\log Z} = 0.25$ dex. Similarly, a gain of $\log_2 10 = 3.32$ bits would add one significant (decimal) digit to the measurement.

For a given observation, the entropy of the posterior distribution of model parameters is $H(\Theta|x)$, which is evaluated over the domain of model parameters, $\Theta$. However, because we do not know [*a priori*]{} what the outcome of an observation will be, we must take the average entropy of the posterior over all possible observations to obtain an accurate measure 