  It is common lore that Parton Distribution Functions (PDFs) in the
$\overline{\rm { MS}}$ factorization scheme can become negative beyond leading
order due to the collinear subtraction which is needed in order to define
partonic cross sections. We show that this is in fact not the case and
next-to-leading order (NLO) $\overline{\rm { MS}}$ PDFs are actually positive
in the perturbative regime. In order to prove this, we modify the subtraction
prescription, and perform the collinear subtraction in such a way that partonic
cross sections remain positive. This defines a factorization scheme in which
PDFs are positive. We then show that positivity of the PDFs is preserved when
transforming from this scheme to $\overline{\rm { MS}}$, provided only the
strong coupling is in the perturbative regime, such that the NLO scheme change
is smaller than the LO term.
