Sometimes size does not matter
  Cosmological fine-tuning has traditionally been associated with the
narrowness of the intervals in which the parameters of the physical models must
be located to make life possible. A more thorough approach focuses on the
probability of the interval, not on its size. Most attempts to measure the
probability of the life-permitting interval for a given parameter rely on a
Bayesian statistical approach for which the prior distribution of the parameter
is uniform. However, the parameters in these models often take values in spaces
of infinite size, so that a uniformity assumption is not possible. This is
known as the normalization problem. This paper explains a framework to measure
tuning that, among others, deals with normalization, assuming that the prior
distribution belongs to a class of maximum entropy (maxent) distributions. By
analyzing an upper bound of the tuning probability for this class of
distributions the method solves the so-called weak anthropic principle, and
offer a solution, at least in this context, to the well-known lack of
invariance of maxent distributions. The implication of this approach is that,
since all mathematical models need parameters, tuning is not only a question of
natural science, but also a problem of mathematical modeling. Cosmological
tuning is thus a particular instantiation of a more general scenario.
Therefore, whenever a mathematical model is used to describe nature, not only
in physics but in all of science, tuning is present. And the question of
whether the tuning is fine or coarse for a given parameter -- if the interval
in which the parameter is located has low or high probability, respectively --
depends crucially not only on the interval but also on the assumed class of
prior distributions. Novel upper bounds for tuning probabilities are presented.
