  Axions arise in many theoretical extensions of the Standard Model of particle
physics, in particular the "string axiverse". If the axion masses, $m_a$, and
(effective) decay constants, $f_a$, lie in specific ranges, then axions
contribute to the cosmological dark matter and dark energy densities. We
compute the background cosmological (quasi-)observables for models with a large
number of axion fields, $n_{\rm ax}\sim \mathcal{O}(10-100)$, with the masses
and decay constants drawn from statistical distributions. This reduces the
number of parameters from $2n_{\rm ax}$ to a small number of "hyperparameters".
We consider a number of distributions, from those motivated purely by
statistical considerations, to those where the structure is specified according
to a class of M-theory models. Using Bayesian methods we are able to constrain
the hyperparameters of the distributions. In some cases the hyperparameters can
be related to string theory, e.g. constraining the number ratio of axions to
moduli, or the typical decay constant scale needed to provide the correct relic
densities. Our methodology incorporates the use of both random matrix theory
and Bayesian networks.
