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We reconstruct the expansion history of the Universe using type Ia supernovae (SN Ia) in a manner independent of any cosmological model assumptions. To do so, we implement a non-parametric iterative smoothing method on the Joint Light-curve Analysis (JLA) data while exploring the SN Ia light-curve hyper-parameter space by Monte Carlo Markov Chain sampling. We test to see how the posteriors of these hyper-parameters depend on cosmology, whether using different dark energy models or reconstructions shift these posteriors. Our constraints on the SN Ia light-curve hyper-parameters from our model-independent analysis are very consistent with the constraints from using different parametrizations of the equation of state of dark energy, namely the flat $Λ\mathrm{CDM}$ cosmology, the Chevallier-Polarski-Linder (CPL) model, and the Phenomenologically Emergent Dark Energy (PEDE) model. This implies that the distance moduli constructed from the JLA data are mostly independent of the cosmological models. We also studied that the possibility the light-curve parameters evolve with redshift and our results show consistency with no evolution. The reconstructed expansion history of the Universe and dark energy properties also seem to be in good agreement with the expectations of the standard $Λ\mathrm{CDM}$ model. However, our results also indicate that the data still allow for considerable flexibility in the expansion history of the Universe.