  Young supernova remnants (SNRs) shocks are believed to be the main sites of
galactic cosmic rays production, showing X-ray synchrotron dominated spectra in
the vicinity of their shock. While a faint thermal signature left by the
shocked interstellar medium (ISM) should also be found in the spectra, proofs
for such an emission in Tycho's SNR have been lacking. We perform an extended
statistical analysis of the X-ray spectra of five regions behind the blast wave
of Tycho's SNR using \textit{Chandra} archival data. We use Bayesian inference
to perform extended parameter space exploration and sample the posterior
distributions of a variety of models of interest. According to Bayes factors,
spectra of all five regions of analysis are best described by composite
three-component models taking into account non-thermal emission, ejecta
emission and shocked ISM emission. The shocked ISM stands out the most in the
Northern limb of the SNR. We find for the shocked ISM a mean electron
temperature $kT_{\rm{e}}=0.96^{+1.33}_{-0.51}$ keV for all regions and a mean
ionization timescale $n_{\rm e}t=2.55^{+0.5}_{-1.22}\times10^9$ cm$^{-3}$s
resulting in a mean ambient density $n_{\rm e}=0.32^{+0.23}_{-0.15}$ cm$^{-3}$
around the remnant. We performed an extended analysis of the Northern limb, and
show that the measured synchrotron cutoff energy is not well constrained in the
presence of a shocked ISM component. Such results cannot currently be further
investigated by analysing emission lines in the 0.5-1 keV range, because of the
low \textit{Chandra} spectral resolution in this band. We show with simulated
spectra that X-IFU future performances will be crucial to address this point.
