  We present a new approach to measure the power-law temperature density
relationship $T=T_0 (\rho / \bar{\rho})^{\gamma -1}$ and the UV background
photoionization rate $\Gamma_{\rm HI}$ of the IGM based on the Voigt profile
decomposition of the Ly$\alpha$ forest into a set of discrete absorption lines
with Doppler parameter $b$ and the neutral hydrogen column density $N_{\rm
HI}$. Previous work demonstrated that the shape of the $b$-$N_{\rm HI}$
distribution is sensitive to the IGM thermal parameters $T_0$ and $\gamma$,
whereas our new inference algorithm also takes into account the normalization
of the distribution, i.e. the line-density d$N$/d$z$, and we demonstrate that
precise constraints can also be obtained on $\Gamma_{\rm HI}$. We use
density-estimation likelihood-free inference (DELFI) to emulate the dependence
of the $b$-$N_{\rm HI}$ distribution on IGM parameters trained on an ensemble
of 624 Nyx hydrodynamical simulations at $z = 0.1$, which we combine with a
Gaussian process emulator of the normalization. To demonstrate the efficacy of
this approach, we generate hundreds of realizations of realistic mock HST/COS
datasets, each comprising 34 quasar sightlines, and forward model the noise and
resolution to match the real data. We use this large ensemble of mocks to
extensively test our inference and empirically demonstrate that our posterior
distributions are robust. Our analysis shows that by applying our new approach
to existing Ly$\alpha$ forest spectra at $z\simeq 0.1$, one can measure the
thermal and ionization state of the IGM with very high precision ($\sigma_{\log
T_0} \sim 0.08$ dex, $\sigma_\gamma \sim 0.06$, and $\sigma_{\log \Gamma_{\rm
HI}} \sim 0.07$ dex).
