Bayesian statisticians often talk about models “learning” as data accumulate. Here’s an example that applies information theory to quantify how much you can learn from an experiment using the same likelihood function but two different priors: a conjugate prior and a robust prior.
Here’s an example from a paper Luis Pericchi and I wrote recently. Suppose X ~ Normal(θ, 1) where the prior on θ is either a standard Cauchy distribution or a normal distribution with mean 0 and variance 2.19. (The variance on the normal was chosen following an example by Jim Berger so that both priors put half their mass on the interval [-1, 1].)
The expected information gain from a single observation using the normal (conjugate) prior was 0.58. The corresponding gain for the Cauchy (robust) prior was 1.20. Because robust priors are more responsive to data, the expected gain in information is larger (in this case twice as large) when using these priors.
Related: Quantifying information content