  We consider the information that can be derived about massive black-hole
binary populations and their formation history solely from current and possible
future pulsar timing array (PTA) results. We use models of the stochastic
gravitational-wave background from circular massive black hole binaries with
chirp mass in the range $10^6 - 10^{11} M_\odot$ evolving solely due to
radiation reaction. Our parameterised models for the black hole merger history
make only weak assumptions about the properties of the black holes merging over
cosmic time. We show that current PTA results place an upper limit on the black
hole merger density which does not depend on the choice of a particular merger
history model, however they provide no information about the redshift or mass
distribution. We show that even in the case of a detection resulting from a
factor of 10 increase in amplitude sensitivity, PTAs will only put weak
constraints on the source merger density as a function of mass, and will not
provide any additional information on the redshift distribution. Without
additional assumptions or information from other observations, a detection
cannot meaningfully bound the massive black hole merger rate above zero or any
particular mass.
