  Direct detection of gravitational waves by pulsar timing arrays will become
feasible over the next few years. In the low frequency regime ($10^{-7}$ Hz --
$10^{-9}$ Hz), we expect that a superposition of gravitational waves from many
sources will manifest itself as an isotropic stochastic gravitational wave
background. Currently, a number of techniques exist to detect such a signal;
however, many detection methods are computationally challenging. Here we
introduce an approximation to the full likelihood function for a pulsar timing
array that results in computational savings proportional to the square of the
number of pulsars in the array. Through a series of simulations we show that
the approximate likelihood function reproduces results obtained from the full
likelihood function. We further show, both analytically and through
simulations, that, on average, this approximate likelihood function gives
unbiased parameter estimates for astrophysically realistic stochastic
background amplitudes.
