  The analysis method currently proposed to search for isotropic stochastic
radiation of primordial or astrophysical origin with the Laser Interferometer
Space Antenna (LISA) relies on the combined use of two LISA channels, one of
which is insensitive to gravitational waves, such as the symmetrised Sagnac.
For this method to work, it is essential to know how the instrumental noise
power in the two channels are related to one another; however, no quantitative
estimates of this key information are available to date. The purpose of our
study is to assess the performance of the symmetrised Sagnac method for
different levels of prior information regarding the instrumental noise. We
develop a general approach in the framework of Bayesian inference and an
end-to-end analysis algorithm based on Markov Chain Monte Carlo methods to
compute the posterior probability density functions of the relevant model
parameters. We apply this method to data released as part of the second round
of the Mock LISA Data Challenges. For the selected (and somewhat idealised)
cases considered here, we find that a prior uncertainty of a factor ~2 in the
ratio between the power of the instrumental noise contributions in the two
channels allows for the detection of isotropic stochastic radiation. More
importantly, we provide a framework for more realistic studies of LISA's
performance and development of analysis techniques in the context of searches
for stochastic signals.
