Characterizing the Lyman-$\alpha$ flux probability distribution function
  using Legendre polynomials
  The Lyman-$\alpha$ forest is a highly non-linear field with a lot of
information available in the data beyond the power spectrum. The flux
probability distribution function (PDF) has been used as a successful probe of
small-scale physics. In this paper we argue that measuring coefficients of the
Legendre polyonomial expansion of the PDF offers several advantages over
measuring the binned values as is commonly done. In particular, $n$-th
coefficient can be expressed as a linear combination of the first $n$ moments,
allowing these coefficients to be measured in the presence of noise and
allowing a clear route for marginalisation over mean flux. Moreover, in the
presence of noise, our numerical work shows that a finite number of
coefficients are well measured with a very sharp transition into noise
dominance. This compresses the available information into a small number of
well-measured quantities. We find that the amount of recoverable information is
a very non-linear function of spectral noise that strongly favors fewer quasars
measured at better signal to noise.
