We investigate the possibilities of New Physics affecting the Standard Model (SM) Higgs sector. An effective Lagrangian with dimension-six operators is used to capture the effect of New Physics. We carry out a global Bayesian inference analysis, considering the recent LHC data set including all available correlations, as well as results from Tevatron. Trilinear gauge boson couplings and electroweak precision observables are also taken into account. The case of weak bosons tensorial couplings is closely examined and NLO QCD corrections are taken into account in the deviations we predict. We consider two scenarios, one where the coefficients of all the dimension-six operators are essentially unconstrained, and one where a certain subset is loop suppressed. In both scenarios, we find that large deviations from some of the SM Higgs couplings can still be present, assuming New Physics arising at 3 TeV. In particular, we find that a significantly reduced coupling of the Higgs to the top quark is possible and slightly favored by searches on Higgs production in association with top quark pairs. The total width of the Higgs boson is only weakly constrained and can vary between 0.7 and 2.7 times the Standard Model value within 95% Bayesian credible interval (BCI). We also observe sizeable effects induced by New Physics contributions to tensorial couplings. In particular, the Higgs boson decay width into $Z\gamma$ can be enhanced by up to a factor 12 within 95% BCI.