**Author:** Maria Dainotti

**Abstract:** We construct a fully Bayesian methodology tuned to the inference of the dark energy equation of state as a function of redshift in a completely non-parametric way, without imposing any a priori w(z) functional form. As a check of the method, we test the complete scheme through the extensive use of synthetic data sets produced from a known cosmological model, and having the same relative errors and redshift distribution as the real data. This allows to confirm the accurate w(z) reconstruction capabilities of the method proposed. Through an updated luminosity time LX - Ta correlation for GRBs X-ray afterglows (known also as Dainotti et al. correlation), we are able to use a large GRB sample from the Swift satellite asaccurate tracers of the expansion history of the Universe out to z = 8.2. Within the assumption of a GR universe with an arbitrary dark energy term, the maximum likelihood solution combining supernovae type Ia, our GRB sample and BAO observations is extremely close to a constant w = -1 solution out to z = 8.2, although variations of ± 0.3 cannot be ruled out. If one imposes the restriction of a constant w, we obtain w = -1(+0.06 -0.04), highly consistent with a cosmological constant, firmly determined out to z = 8.2. As additional constraints in the low redshift range, the method yields highly accurate maximum likelihood inferences for the present day parameters: H_{0} = 69.5 ± 0.3kms^{-1}Mpc^{-1} and Ω_{Λ,0} = 0.72 ± 0.015.