Bayesian calibration of the mixing length parameter $\alpha_{ML}$ and of
  the helium-to-metal enrichment ratio $\Delta Y/\Delta Z$ with open clusters:
  the Hyades test-bed
  We tested the capability of a Bayesian procedure to calibrate both the helium
abundance and the mixing length parameter ($\alpha_{ML}$), using precise
photometric data for main-sequence (MS) stars in a cluster with negligible
reddening and well-determined distance. The method has been applied first to a
mock data set generated to mimic Hyades MS stars and then to the real Hyades
cluster. We tested the impact on the results of varying the number of stars in
the sample, the photometric errors, and the estimated [Fe/H]. The analysis of
the synthetic data set shows that $\alpha_{ML}$ is recovered with a very good
precision in all the analysed cases (with an error of few percent), while
[Fe/H] and the helium-to-metal enrichment ratio $\Delta Y/\Delta Z$ are more
problematic. If spectroscopic determinations of [Fe/H] are not available and
thus [Fe/H] has to be recovered alongside with $\Delta Y/\Delta Z$ and
$\alpha_{ML}$, the well-known degeneracy between [Fe/H]-$\Delta Y/\Delta
Z$-$\alpha_{ML}$ could result in a large uncertainty on the recovered
parameters, depending on the portion of the MS used for the analysis. On the
other hand, the prior knowledge of an accurate [Fe/H] value puts a strong
constraint on the models, leading to a more precise parameters recovery. Using
the current set of PISA models, the most recent [Fe/H] value and the Gaia
photometry and parallaxes for the Hyades cluster, we obtained the average
values $<\alpha_{ML}>=2.01\pm0.05$ and $<\Delta Y/\Delta Z>=2.03\pm0.33$,
sensitively reducing the uncertainty in these important parameters.
