Observations of the Lyman-$\alpha $ (Ly$\alpha $ ) forest from spectroscopic
surveys such as BOSS/eBOSS, or the ongoing DESI, offer a unique window to study
the growth of structure on megaparsec scales. Interpretation of these
measurements is a complicated task, requiring hydrodynamical simulations to
model and marginalise over the thermal and ionisation state of the
intergalactic medium. This complexity has limited the use of Ly$\alpha $
clustering measurements in joint cosmological analyses. In this work we show
that the cosmological information content of the 1D power spectrum
(${P}_{1\mathrm{D}}$ ) of the Ly$\alpha $ forest can be compressed into a simple
two-parameter likelihood without any significant loss of constraining power. We
simulate ${P}_{1\mathrm{D}}$ measurements from DESI using hydrodynamical
simulations and show that the compressed likelihood is model independent and
lossless, recovering unbiased results even in the presence of massive neutrinos
or running of the primordial power spectrum.