  We present a semi-analytic model for the Lyman-$\alpha$ forest that is
inspired by the Halo Model. This model is built on the absorption line
decomposition of the forest. Flux correlations are decomposed into those within
each absorption line (the 1-absorber term) and those between separate lines
(the 2-absorber term), treating the lines as biased tracers of the underlying
matter fluctuations. While the nonlinear exponential mapping between optical
depth and flux requires an infinite series of moments to calculate any
statistic, we show that this series can be resumed (truncating at the desired
order in the linear matter overdensity). We focus on the $z=2-3$ line-of-sight
power spectrum. Our model finds that 1-absorber term dominates the power on all
scales, with most of its contribution coming from HI columns of
$10^{14}-10^{15}\;\mathrm{cm^{-2}}$, while the smaller 2-absorber contribution
comes from lower columns that trace overdensities of a few. The prominence of
the 1-absorber correlations indicates that the line-of-sight power spectrum is
shaped principally by the lines' number densities and their absorption
profiles, with correlations between lines contributing to a lesser extent. We
present intuitive formulae for the effective optical depth as well as the
large-scale limits of 1-absorber and 2-absorber terms, which simplify to
integrals over the HI column density distribution with different
equivalent-width weightings. With minimalist models for the bias of absorption
systems and their peculiar velocity broadening, our model predicts values for
the density bias and velocity gradient bias that are consistent with those
found in simulations.
