  Inference from limited data requires a notion of measure on parameter space,
most explicit in the Bayesian framework as a prior. Here we demonstrate that
Jeffreys prior, the best-known uninformative choice, introduces enormous bias
when applied to typical scientific models. Such models have a relevant
effective dimensionality much smaller than the number of microscopic
parameters. Because Jeffreys prior treats all microscopic parameters equally,
it is from uniform when projected onto the sub-space of relevant parameters,
due to variations in the local co-volume of irrelevant directions. We present
results on a principled choice of measure which avoids this issue, leading to
unbiased inference in complex models. This optimal prior depends on the
quantity of data to be gathered, and approaches Jeffreys prior in the
asymptotic limit. However, this limit cannot be justified without an impossibly
large amount of data, exponential in the number of microscopic parameters.
