  We consider the inverse problem in pulsar timing array (PTA) analysis,
investigating what astrophysical information about the underlying massive black
hole binary (MBHB) population can be recovered from the detection of a
stochastic gravitational wave background (GWB). We employ a physically
motivated model that connects the GWB spectrum to a series of parameters
describing the underlying redshift evolution of the MBHB mass function and to
the typical eccentricity they acquire while interacting with the dense
environment of post merger galactic nuclei. This allows the folding in of
information about the spectral shape of the GWB into the analysis. The priors
on the model parameters are assumed to be uninformative and consistent with the
current lack of secure observations of sub-parsec MBHBs. We explore the
implications of current upper limits as well as of future detections with a
variety of PTA configurations. We confirm our previous finding that current
upper limits can only place an upper bound on the overall MBHB merger rate.
Depending on the properties of the array, future detections can also constrain
several MBHB population models at different degrees of fidelity. In particular,
a simultaneous detection of a steepening of the spectrum at high frequency and
a bending at low frequency will place strong constraints on both the MBHB mass
function and on the typical eccentricity of inspiralling MBHBs, providing
insights on MBHB astrophysics unlikely to be achievable by any other means.
