  We use the language of uninformative Bayesian prior choice to study the
selection of appropriately simple effective models. We advocate for the prior
which maximizes the mutual information between parameters and predictions,
learning as much as possible from limited data. When many parameters are poorly
constrained by the available data, we find that this prior puts weight only on
boundaries of the parameter manifold. Thus it selects a lower-dimensional
effective theory in a principled way, ignoring irrelevant parameter directions.
In the limit where there is sufficient data to tightly constrain any number of
parameters, this reduces to Jeffreys prior. But we argue that this limit is
pathological when applied to the hyper-ribbon parameter manifolds generic in
science, because it leads to dramatic dependence on effects invisible to
experiment.
