  The thermal state of the intergalactic medium (IGM) contains vital
information about the epoch of reionization, one of the most transformative yet
poorly understood periods in the young universe. This thermal state is encoded
in the small-scale structure of Lyman-$\alpha$ (Ly$\alpha$) absorption in
quasar spectra. The 1D flux power spectrum measures the average small-scale
structure along quasar sightlines. At high redshifts, where the opacity is
large, averaging mixes high signal-to-noise ratio transmission spikes with
noisy absorption troughs. Wavelet amplitudes are an alternate statistic that
maintains spatial information while quantifying fluctuations at the same
spatial frequencies as the power spectrum, giving them the potential to more
sensitively measure the small-scale structure. Previous Ly$\alpha$ forest
studies using wavelet amplitude probability density functions (PDFs) used
limited spatial frequencies and neglected strong correlations between PDF bins
and across wavelets scales, resulting in sub-optimal and unreliable parameter
inference. Here we present a novel method for performing statistical inference
using wavelet amplitude PDFs that spans the full range of spatial frequencies
probed by the power spectrum and that fully accounts for these correlations. We
applied this procedure to realistic mock data drawn from a simple thermal model
parameterized by the temperature at mean density, $T_0$, and find that wavelets
deliver 1$\sigma$ constraints on $T_0$ that are on average 7% more sensitive at
$z=5$ (12% at $z=6$) than those from the power spectrum. We consider the
possibility of combing wavelet PDFs with the power, but find that this does not
lead to improved sensitivity.
