produce the narrowest constraints on the model parameters. However, this is more complicated than finding the smallest error bars. The error bars are not necessarily informative, for example, if we attempt to use them to characterize a multi-peaked distribution, and they are less helpful when we also wish to quantify the observing time needed to reach a certain level of precision. To do this, we must have a quantitative measure of the amount of information provided by an observation.

An observation $x$ with probability $p(x)$ carries an information content of $I = -\log_2 p(x)$ bits [@shannon48]. In other words, a minimum of $I$ bits are required to encode the value of $x$. We care about this because it effectively tells us how much information we know about $x$, where $x$ in this case may be the observed value of some property of the planet. The entropy, $H(X)$, of the complete probability distribution $p(x)$, per @shannon48, is the expectation value of $I$: $$H(X) = -\sum_{x\in X} p(x)\log_2 p(x).$$ This represents the average information needed to encode a value of $x$ from the domain $X$, which is the set of all possible observed values of $x$. For a continuous distribution, this becomes[^2] 