  One of the sources of gravitational waves for the proposed space-based
gravitational wave detector, the Laser Interferometer Space Antenna (LISA), are
the inspirals of compact objects into supermassive black holes in the centres
of galaxies - extreme-mass-ratio inspirals (EMRIs). Using LISA observations, we
will be able to measure the parameters of each EMRI system detected to very
high precision. However, the statistics of the set of EMRI events observed by
LISA will be more important in constraining astrophysical models than extremely
precise measurements for individual systems. The black holes to which LISA is
most sensitive are in a mass range that is difficult to probe using other
techniques, so LISA provides an almost unique window onto these objects. In
this paper we explore, using Bayesian techniques, the constraints that LISA
EMRI observations can place on the mass function of black holes at low
redshift. We describe a general framework for approaching inference of this
type --- using multiple observations in combination to constrain a
parameterised source population. Assuming that the scaling of EMRI rate with
black hole mass is known and taking a black hole distribution given by a simple
power law, dn/d(ln M) = A (M/M_*)^b, we find that LISA could measure the
parameters to a precision of D(ln A) ~ 0.08, and D(b) ~ 0.03 for a reference
model that predicts ~1000 events. Even with as few as 10 events, LISA should
constrain the slope to a precision ~0.3, which is the current level of
observational uncertainty in the low-mass slope of the black hole mass
function. We also consider a model in which A and b evolve with redshift, but
find that EMRI observations alone do not have much power to probe such an
evolution.
