(\[chi2\_salt2\_new\])\].

The $\chi^2$ approach from MLCS2k2 output
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The MLCS2k2 light curve fitter is also a distance estimator and gives us directly a cosmology-independent estimation of the distance modulus. In this context, the analogue of Eq. (\[chi2\_salt2\]) is $$\chi^2_{\mathrm{MLCS2k2}}(\boldsymbol{\theta},h) = \sum_i^N
\frac{[\mu_{i}-\mu_\mathrm{th}(z_i;\boldsymbol{\theta},h)]^2}{\sigma_i^2 +
\sigma_\mathrm{int}^ 2+\sigma^2_{\mu,z,i}}
\label{chi2_mlcs2k2}$$ where $\sigma_i$ is the distance modulus dispersion as given by MLCS2k2.

The procedure to obtain $\sigma_\mathrm{int}$ is similar to the one described in the previous Subsection, however, in this case we use only a subsample of nearby SNe Ia and not the full one, as for the SALT2 analysis. After setting up the value of $\sigma_\mathrm{int}$, we minimize the $\chi^2_{\mathrm{MLCS2k2}}$ using the full SNe Ia sample to obtain the best fit values for $\boldsymbol{\theta}$ and $h$.

The proposed likelihood approach {#sec:likelihood}
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Considering the SNe Ia light curve fitting parameters as Gaussian distributed random variables, we propose to take as starting point the likelihood $$L=\frac{1}{\sqrt{(2\pi)^N\hbox{det}\,\boldsymbol{\Sigma}}}\exp(-\boldsymbol{X}^T
\boldsymbol{\Sigma}^{-1}\boldsymbol{X}/2)\,,
\label{like}$$ which is related to the $\chi^2$ in Eq. (\[chi2\_base\]) by $$\mathcal{L}:=-2\ln L=\chi^2  + \ln
\hbox{det}\,\boldsymbol{\Sigma}+ N\ln (2\pi)\,.
\label{loglike}$$ Eqs. (\[like\]) and (\[loglike\]) are the single basis upon which our whole statistical procedure lies. When the full covariance of the problem is known, minimizing $\chi^2$ is completely equivalent 