We apply two Bayesian hierarchical inference schemes to infer shear power spectra, shear maps and cosmological parameters from the Canada–France–Hawaii Telescope (CFHTLenS) weak lensing survey – the first application of this method to data. In the first approach, we sample the joint posterior distribution of the shear maps and power spectra by Gibbs sampling, with minimal model assumptions. In the second approach, we sample the joint posterior of the shear maps and cosmological parameters, providing a new, accurate and principled approach to cosmological parameter inference from cosmic shear data. As a first demonstration on data, we perform a two-bin tomographic analysis to constrain cosmological parameters and investigate the possibility of photometric redshift bias in the CFHTLenS data. Under the baseline ΛCDM (Λ cold dark matter) model, we constrain
S8=σ8(Ωm/0.3)0.5=0.67+0.03−0.03
S8=σ8(Ωm/0.3)0.5=0.67−0.03+0.03

(68 per cent), consistent with previous CFHTLenS analyses but in tension with Planck. Adding neutrino mass as a free parameter, we are able to constrain ∑mν < 4.6 eV (95 per cent) using CFHTLenS data alone. Including a linear redshift-dependent photo-z bias Δz = p2(z − p1), we find
p1=−0.25+0.53−0.60
p1=−0.25−0.60+0.53

and
p2=−0.15+0.17−0.15
p2=−0.15−0.15+0.17

, and tension with Planck is only alleviated under very conservative prior assumptions. Neither the non-minimal neutrino mass nor photo-z bias models are significantly preferred by the CFHTLenS (two-bin tomography) data.