  We study the correlations of the shear signal between triplets of sources in
the Canada-France-Hawaii Lensing Survey (CFHTLenS) to probe cosmological
parameters via the matter bispectrum. In contrast to previous studies, we
adopted a non-Gaussian model of the data likelihood which is supported by our
simulations of the survey. We find that for state-of-the-art surveys, similar
to CFHTLenS, a Gaussian likelihood analysis is a reasonable approximation,
albeit small differences in the parameter constraints are already visible. For
future surveys we expect that a Gaussian model becomes inaccurate. Our
algorithm for a refined non-Gaussian analysis and data compression is then of
great utility especially because it is not much more elaborate if simulated
data are available. Applying this algorithm to the third-order correlations of
shear alone in a blind analysis, we find a good agreement with the standard
cosmological model: $\Sigma_8$=$\sigma_8$ $(\Omega_{\rm
m}/0.27)^{0.64}$=$0.79^{+0.08}_{-0.11}$ for a flat $\Lambda\rm CDM$ cosmology
with $h=0.7\pm0.04$ ($68\%$ credible interval). Nevertheless our models provide
only moderately good fits as indicated by $\chi^2/{\rm dof}=2.9$, including a
$20\%$ r.m.s. uncertainty in the predicted signal amplitude. The models cannot
explain a signal drop on scales around 15 arcmin, which may be caused by
systematics. It is unclear whether the discrepancy can be fully explained by
residual PSF systematics of which we find evidence at least on scales of a few
arcmin. Therefore we need a better understanding of higher-order correlations
of cosmic shear and their systematics to confidently apply them as cosmological
probes.
