We start by discussing different aspects related to the parameterization
of the PDFs at in input scale $Q_0$: the choice of functional form, the theoretical
constraints from the momentum and valence sum rules and the PDF positivity,
as well as the various quark flavour assumptions used in the PDF fits.

\subsubsection{Choice of functional form}\label{sec:fitmeth.PDFpara.func}
%
In order to extract the PDFs, a given choice for its parameterisation in $x$ at the
input scale $Q_0$ must be assumed, which can then be fit to the available data.
%
As described in Sect.~\ref{subsec:dglap}, given the PDFs at some reference scale $Q_0$, the
DGLAP evolution equations can be used to subsequently determine the PDFs at any other scale $Q$.
%
Thus the PDFs are typically parameterised at a low scale $Q_0^2 \sim 1-2\,{\rm GeV}^2$, which can then be evolved up to the scale relevant to e.g. LHC phenomenology.
%
These parametrizations usually adopt the generic form
\be\label{pdffunc}
xf(x,Q^2_0) = A_{f} x^{a_{f}}(1-x)^{b_{f}}I_{f}(x)\;.
\ee
The $(1-x)^{b_{f}}$ term, with $b_{f}>0$, ensures that the PDFs vanish in the elastic $x\to 1$ limit, as we would expect on basic physical grounds.
%
Such a form is also expected from the quark counting rules~\cite{Brodsky:1973kr}
(see also the discussion on~\cite{Ball:2016spl}).
%
There, in this elastic limit all the momentum is carried by the struck parton and the remaining $n_s$ quark become spectators.
%
An analysis of the scaling behaviour for elastic scattering then predicts $b_f=2n_s-1$, that is $b_f=3,5$ and 7 for the valence, sea and gluon PDFs, respectively.

On the other hand, the $x^{a_{f}}$ form dominates at low $x$; in this region, the PDFs might
be  related to the high energy parton--proton scattering amplitudes, which may be calculated using the tools of Regge theory.
%
This scenario predicts such a simple power--like form, with the precise value of the power $a_{f}$ being related to the leading Regge trajectory that is exchanged; for non--singlet distributions (e.g. the valence quarks) this predicts $a_f\sim 0.5$ and for singlet distributions (e.g. the gluon and the sea) this predicts $a_f\sim 0$.

We emphasize that the above discussion only corresponds to quite
general expectations (as opposed to direct
QCD predictions), which do not for example account for the scale dependence of the PDFs.
%
Thus while the high and low $x$ form of
Eq.~(\ref{pdffunc}) is usually adopted, in modern fits the values of the powers themselves are more generally left free where there is sufficient data to constrain them.
%
Interestingly, the comparison of the effective asymptotic large-$x$ exponents
of modern fits with these model predictions~\cite{Ball:2016spl} are consistent
with the quark counting rules expectation.

The $I_{f}(x)$ in Eq.~(\ref{pdffunc}) is the interpolating function, which determines the behaviour of the PDFs away from the $x\to 0$ and 1 limits, where it tends to a constant value.
%
This is assumed to be a smoothly varying function of $x$, for which a variety of choices have been made in the literature. The simplest ansatz, which has been very widely used, is to take a basic polynomial form in $x$ (or $\sqrt{x}$), such as
\be\label{Ipower}
I_{f}(x)=1+c_{f}\sqrt{x}+d_{f}x+...\;.
\ee
Forms of this type are for example taken by the CJ and HERAPDF groups as well as in the
MSTW08 analysis.
%
A similar approach, but where the polynomial enters as the exponent of a power of $x$ or a simple exponential function, are taken by ABM and earlier CT sets, respectively. 

Such a choice is appropriate for a relatively small number of parameters, say only two or three
in addition to $a_f$ and $b_f$.
%
However, as the precision and amount of the data included in the fit increases, it becomes essential to allow for an increasingly flexible parameterisation.
%
As discussed in~\cite{Pumplin:2009bb}, simply adding more parameters to (\ref{Ipower}) can quickly run into the technical issues that large coefficients appear, with large cancellations between the terms. This leads to an unstable $\chi^2$ minimisation and implausibly large variations in $x$ in certain regions.
%
This issue may be solved by instead expanding the interpolating function in terms of a basis of suitably chosen functions with the generic form
\be
I_{f}(x)=\sum_{i=1}^n \alpha_{f,i} P_i(y(x))\;,
\ee
where $y(x)$ is some simple function of $x$.
%
Two possible choices for the functions $P_i$ are Chebyshev and Bernstein polynomials, which are used in the MMHT14 and CT14 sets, respectively.
%
These are chosen as each order of the polynomials is strongly peaked at different values of $y$, and hence $x$, significantly reducing the degree of correlation between the terms.
%
In addition, as the order is increased these tend to probe smaller scale variations in $x$, so that the smoothness requirement for $I(x)$ naturally leads to smaller coefficients $\alpha$ at higher $i$.
%
Thus, while formally equivalent to the simply polynomial expansion in Eq.~(\ref{Ipower}), these are much more convenient for fitting as the number of free parameters $n$ is increased. 

An alternative approach is taken by the NNPDF group. Here, the interpolating function is modelled with a multi--layer feed forward neural network (also known as
a {\it perceptron}), see Sect.~\ref{nnpdf} for more details.
%
In practice, this allows for a greatly increased number of free parameters, with the latest default fit having 37 per PDF, that is around an order of magnitude higher than other sets.
%
The form of Eq.~(\ref{pdffunc}) is still assumed, but
these are pre--processing factors that speed up the minimisation procedure but which do not in principle have to be explicitly included.
%
Nonetheless, the study of~\cite{Ball:2016spl} has shown that the NNPDF fit does exhibit high and low $x$ behaviour that is consistent with Eq.~(\ref{pdffunc}), providing further support for such an assumption in the choice of input PDF parametrization.

\subsubsection{Sum rules}\label{sec:fitmeth.PDFpara.sum}

The flavour quantum numbers of the proton, $uud$ with zero strangeness, are
expressed in the three number or valence sum rules,
\begin{align}
\int_0^1 {\rm d}x\,\left[u(x,Q^2)-\overline{u}(x,Q^2)\right]&=2\;,\\
\int_0^1 {\rm d}x\,\left[d(x,Q^2)-\overline{d}(x,Q^2)\right]&=1\;,\\ \label{strangesum}
\int_0^1 {\rm d}x\,\left[s(x,Q^2)-\overline{s}(x,Q^2)\right]&=0\;,
\end{align}
thus for the valence distributions we must have $a_f>0$ for the exponents in Eq.~(\ref{pdffunc}) or these integrals will diverge.
%
In others words, this constraint is consistent
with the well known result that the $xf$ valence distributions vanish
as $x\to 0$.
%
Although not shown explicitly, a similar constraint applies to the heavy quark PDFs as to the strange PDF.
%
In the absence of any intrinsic heavy flavour, these are automatically satisfied.
%
We also note that this sum rules hold at any value of the scale $Q^2$, and indeed they are satisfied
by DGLAP evolution provided they hold at the input parametrization scale.

The second moment of the sum of PDFs must also obey the momentum sum rule
\be
\label{eq:momsumrule}
\int_0^1 {\rm d}x\, x\left(\sum_{f=1}^{n_f}\left[
q_f(x,Q^2)+\overline{q}_f(x,Q^2)\right]+g(x,Q^2)+\cdots\right)=1\;,
\ee
which expresses the simple physical requirement, arising
from energy conservation,
that the total proton momentum must be equal to the sum of its constituents.
%
In Eq.~(\ref{eq:momsumrule}), the dots indicate possible additional
contributions from other partons in the proton, for instance from the
photon PDF $\gamma(x,Q^2)$
that should be accounted for in PDF sets with QED corrections
(see Sect.~\ref{sec:QED.photon}).
%
Eq.~(\ref{eq:momsumrule}) also implies that for non--valence distributions,
the exponent $a_f$ may be negative, but must be greater than -1 to avoid giving a divergent contribution to the momentum sum rule.

The above sum rules provide additional constraints on the input PDFs, and are typically applied to fix certain parameters, for example the overall normalization $A_f$ of the gluon or
of specific quark flavour combinations.
%
Provided these sum rules are satisfied at the input scale, it follows straightforwardly from
the structure
of the DGLAP evolution that they will be satisfied at any other scale $\mu$.
%
For instance, the $g\to q\overline{q}$ splitting can generate no net $q-\overline{q}$ component,
and while the DGLAP evolution reshuffles the momentum carried between the different partons, it does not generate any
violation of momentum conservation.

\subsubsection{Quark flavour assumptions}\label{sec:fitmeth.PDFpara.flavour}

Assuming that there are $n_f$ active quark flavors at the input
parametrization scale $Q_0$, there will be in general $2n_f+1$ PDFs to be parametrized and fitted
from data.
%
Assuming that the heavy quark PDFs are generated perturbatively,
in addition to the gluon, in many cases  the remaining 6 light quarks PDFs parametrized
are not those in the {\it flavour basis}, namely
\be
u,\quad \bar{u},\quad d,\quad \bar{d},\quad s,\quad \bar{s}\;,
\ee
but rather using other  convenient linear combinations, {\it e.g.} the valence $u_V=u-\bar{u}$ and
$d_V=d-\bar{d}$ distributions
are often adopted.
%
To give one example, the MMHT14 analysis  takes as
fitting basis, in addition to the gluon, the following
quark combinations,
\be
u_V,\quad d_V,\quad \overline{d}-\overline{u},\quad s+\overline{s},\quad s-\overline{s},\quad s+\overline{s}+2(u+\overline{u}+d+\overline{d})\;.
\ee
As another example, the NNPDF3.0 fit parametrizes the  PDFs at the input evolution scale
in the so-called {\it evolution basis}, defined as the eigenvectors of the DGLAP
evolution equations (see also Sect.~\ref{subsec:dglap}),
\bea
\Sigma&=& u+\bar{u}+d+\bar{d}+s+\bar{s} \, \nonumber \\
T_3&=& u+\bar{u} - d - \bar{d}  \, ,\nonumber \\
\label{eq:PDFevolutionbasis}
T_8&=& u+\bar{u} + d + \bar{d} -2s-2\bar{s}  \, , \\
V&=& u-\bar{u}+d-\bar{d}+s-\bar{s} \, \nonumber \\
V_3&=& u-\bar{u} - d + \bar{d}  \, ,\nonumber \\
V_8&=& u- \bar{u} + d - \bar{d} -2s + 2\bar{s}  \, ,\nonumber 
\eea
in addition to the gluon PDF.
%
However, as any particular basis can be trivially related to another by a linear transformation,
the resulting physics should not depend on these choices.
%
On the other hand, different flavour assumptions do often lead to different results in regions
with scarce experimental constraints, such as the large-$x$ region.

Historically, the strange quark has been rather
less well determined than the $u$ and $d$ quark PDFs,
and indeed in many earlier fits $s(x,Q^2)$ was not fitted from
data but rather was fixed according to 
\be
s=\overline{s}\propto  \lp \overline{u}+\overline{d} \rp \;.
\ee
Such a choice is still taken in the CJ15 and HERAPDF fits, due to the more restricted data set. With the increase in available data, the total strangeness $s+\overline{s}$ is now freely parameterised in all global fits.
%
While the sum rule Eq.~(\ref{strangesum}) requires there to be no overall strangeness in the proton, at a given $x$ value there is no requirement for the $s_V=s-\overline{s}$ distribution to vanish, and indeed non--perturbative approaches such as the ``meson cloud model''~\cite{Signal:1987gz} predict a non-zero strange asymmetry.
%
However, the strange difference $s-\overline{s}$ is generally quite poorly determined and still broadly consistent with zero within current uncertainties. From the latest global fits, only  MMHT14 and the NNPDF3 family fit the strange difference $s_V$, while for all other sets it is still assumed that $s=\overline{s}$.
%
Note also that at NNLO, even if $s_V$ is set to zero at the initial evolution scale,
a non-zero strangeness (as well as charm and bottom) asymmetry will be generated dynamically by the DGLAP
evolution equations~\cite{Catani:2004nc}.

The above discussion assumes that the charm PDF $c(x,Q^2)$ is 
generated due to perturbative $g\to c\overline{c}$ splittings, in which case it is
completely determined by the DGLAP evolution equations in
terms of the light quark and gluon PDFs above the charm mass $m_c$ (and vanishing for $Q<m_c$).
%
If the charm PDF is instead fitted, the input flavour assumptions need thus to be modified.
%
In the case of NNPDF3.1~\cite{Ball:2016neh}, the evolution basis of Eq.~(\ref{eq:PDFevolutionbasis})
is supplemented with $c^+=c+\bar{c}$, which is also freely parametrized with a 37-parameter
artificial neural
network, while it is assumed that $c^-=c-\bar{c}=0$.
%
This option is also adopted in other recent studies where the charm PDF is fitted,
such as in the CT14 IC analysis~\cite{Hou:2017khm}
(see also Sect.~\ref{sec:structure.charm}).
%
Note that in general different flavour assumptions concerning the parametrized
charm PDF are conceivable,
for instance Eq.~(\ref{eq:PDFevolutionbasis}) could be also generalized by adding
\be
T_{15}= u+\bar{u} + d + \bar{d} +s+\bar{s}-3(c+3\bar{c})  \, , 
\ee
though this option has the drawback that the connection with charm-sensitive
observables is less direct.
%

An important issue related to the PDF parametrization is that of the positivity.
%
While, beyond LO, PDFs are scheme-dependent quantities and thus in principle can become negative,
physical observables such as cross-sections and structure functions should always be positive-definite.
%
This constrain is incorporated in the (N)NLO global fits in different ways.
%
For instance, in CT14 all PDFs are made by construction positive-definite, while MMHT14 allows
the small-$x$ gluon PDF to become negative.
%
In the case of the NNPDF family of fits, no positivity constraints are imposed at the
PDF level (except of course at the LO case),
but during the fit the strict positivity of a range of physical
cross-sections is imposed by means of a Lagrange multiplier.
%
Specifically, in the NNPDF3 sets the positivity of the following cross-sections
is imposed at $Q^2=5$ GeV$^2$: $F_2^u$,$F_2^d$, $F_2^s$, $F_L$,
$\sigma_{\rm DY}^{u\bar{u}}$, $\sigma_{\rm DY}^{d\bar{d}}$, and
$\sigma_{\rm DY}^{s\bar{s}}$.
%
Note that in general this positivity constraint applies to all conceivable cross-sections,
including for instance those that involve hypothetical new particles, and is not
restricted to the actual cross-sections that are accessible experimentally.

