Individual PDF sets from different groups are widely used when comparing precision
theoretical predictions with LHC measurements and in the assessment of the accuracy
of PDF sets themselves.
%
However, for several LHC applications an assessment
of the {\it total} PDF uncertainty for certain observables, by taking into account 
predictions from all applicable PDF sets, is advantageous.
%
This will for example be the case in the
extraction of the couplings of the Higgs boson from the
experimental cross-sections, or the calculation of signal and
background rates in searches for BSM physics.

For such purposes, a statistical procedure is needed with which to combine the
results from different PDF sets.
%
However, a statistical combination of PDF sets is rather more complicated than
in the case of for example the world average of the strong coupling constant or heavy--quark masses,
since it combines functions which have in principle an infinite number of degrees of freedom.
%
In addition, a suitable prescription must also
accommodate the fact that the individual PDF sets
are not identical either in their central values or in their uncertainties, and it
should account for possible correlations between PDF sets from different
groups.
%
Moreover, it would be desirable to achieve a final representation of this
combined PDF set in terms of a relatively small number of Hessian eigenvectors
or Monte Carlo replicas.

The PDF4LHC Working Group 2010
recommendation proposed the use of a simple envelope prescription~\cite{Alekhin:2011sk,
Ball:2012wy,Botje:2011sn}.
%
That is, the PDF determinations from different groups are treated as
instances of a probability distribution affected by unknown sources of systematics,
rather than 
statistically distributed instances of an underlying probability distribution.
%
This envelope prescription can also only be applied at the level of individual observables,
thereby losing the information on PDF-induced correlations.
%
Given the better understanding of current PDF determinations, the
relatively good agreement between the current releases of the
three main global PDF sets,
and the high precision demands for LHC Run II studies, such a
prescription is therefore certainly inadequate. 

With the above considerations in mind, the updated 2015
PDF4LHC recommendation was proposed as a replacement~\cite{Butterworth:2015oua}.
%
There, a number of
criteria for the individual PDF sets to be considered for the combination were
adopted.
%
First of all,
the individual PDF sets should be based on a global determination with a large number
of datasets from a variety of experiments, that is,
DIS and hadron--hadron scattering in
fixed--target and collider experiments.
%
Second, the hard cross sections for DIS and hadron--hadron
scattering processes used in the extraction should be evaluated up to two loops in
QCD in a GM-VFN scheme, with a maximum number of $n_f=5$ active quark flavors.
%
Third, all known
experimental and procedural sources of uncertainties should be properly accounted
for, including the experimental uncertainties propagated from data, uncertainties
due to the incompatibility of different data sets, and uncertainties due to the functional
form of PDFs.

It was also decided that the combination should be carried out with
a central value of $\alpha_s(m_Z)=0.118$ at both NNLO and NLO,  with the corresponding
total uncertainty taken to be $\delta \alpha_s(m_Z)= 0.0015$, consistent with the most
updated PDG world-average~\cite{Beringer:1900zz}.
%
The heavy quark masses used in individual PDF sets
are not currently required to be the same\footnote{It would be desirable in the
future for all PDF groups provide error sets with common choices of heavy-quark masses and
furthermore to include the uncertainties due to the mass inputs, similarly to the case of $\alpha_s$.}
but should be compatible with their
world--average values.
%
The existing PDF sets satisfying all of the above requirements at
present have been identified as CT14~\cite{Dulat:2015mca}, MMHT2014~\cite{Harland-Lang:2014zoa},
and NNPDF3.0~\cite{Ball:2014uwa}.
%
The PDF4LHC 2015 PDF
sets are therefore statistical combination of these three global analyses, although the method
can be straightforwardly generalized to include additional PDF sets satisfying the
same quality conditions.

An important point here is that this
statistical combination can only be carried out efficiently using the Monte Carlo method,
as different PDF determinations adopt different forms for the PDF parametrizations.
In the first step, the CT14 and MMHT2014 PDFs, which are originally in their Hessian form, are
converted into their Monte Carlo representations by applying the Watt--Thorne method
with symmetric formula~\cite{Watt:2012tq}.
%
It has been validated that a MC ensemble with
$N_{\rm rep}=300$ replicas is sufficient to reproduce the central value and uncertainties of the original
Hessian PDFs to high precision.
%
The NNPDF3.0 PDF set is already in a Monte Carlo
form with 1000 replicas. Following the idea of individual PDF determinations
as equally likely representations of an underlying probability distribution, a
combined PDF set is then build by taking 300 MC replicas from each input PDF sets and
merging them equally.

The resulting combined PDF set, an ensemble of 900 MC replicas,
is referred to as the MC900 or PDF4LHC15\_prior, and represents the combined 
probability distribution of the PDFs.
%
However, such a set of 900 PDFs would be unmanageably large for most applications, in particular 
given the time and storage cost required for complicated NNLO calculations and
experimental simulations.
%
Therefore, various methods have been developed to reduce the size
of the combined sets, while minimizing the information loss according to various statistical measures.

The first method applies the META--PDFs framework~\cite{Gao:2013bia}. Here, a flexible
functional form with Bernstein polynomials is chosen to parametrize the PDFs at an
initial scale. Each replica in the MC900 ensemble is then represented by a group
of PDF parameters through a fit to the chosen parametrizations, by minimizing
a metric function.
%
The prior probability distribution of PDFs is thus transformed
into the probability distributions in the PDF parameter space.
%
The covariance matrix of
the PDF parameters is calculated as usual,
\begin{equation}
{\rm cov}(a_l,a_m)=\frac{1}{N_{\rm rep}-1}\sum_{k=1}^{N_{\rm rep}}(a_l^{(k)}-a_l^{(0)})(a_m^{(k)}-a_m^{(0)}) \, ,
\end{equation}
where $a_l^{(0)}$ and $a_l^{(k)}$ denote the fitted PDF parameters from the central set and
the $k$-th MC replica respectively,
and $N_{\rm rep}$ is the total number of MC replicas.
%
The covariance matrix can be diagonalized by an orthogonal transformation.
Eigenvectors are then calculated and ordered according to their impact on the PDF uncertainties
with a designed error metric; the eigenvectors with smaller contributions can be dropped according to the
accuracy required.

In the final step of the META-PDF procedure,
a central PDF set and a group of orthogonal error PDF sets
are generated under the assumption of a multi--Gaussian distribution, which
 can then be used in a
similar way as the conventional Hessian PDF sets.
%
For example, the 68\% CL uncertainty or
$1\sigma$ error on a generic cross-section $\mathcal{F}$ is given by
\begin{equation}
\label{eq:syhe}
\delta^{\rm PDF} \mathcal{F}=\sqrt{\sum_{i=1}^{N_{\rm eig}}(\mathcal{F}_{i}-\mathcal{F}_0)^2}\, ,
\end{equation}
where $\mathcal{F}_0$ is the prediction on observable $X$ given by the central set and
$\mathcal{F}_i$ is the prediction given by the $i$-th error set.
%
Note there is only one
error set along each eigenvector/orthogonal direction since
symmetric Gaussian distributions are assumed in this case by construction. 

The second method that can be used to compress the MC900 prior set
is the {\tt mc2h} algorithm with Singular Value Decomposition,
followed by the Principle Component Analysis~\cite{Carrazza:2015aoa}.
%
The idea underlying this method is to first
discretize the PDFs at $N_xN_{\rm pdf}$ grid nodes, where $N_x$ denotes the total number of grid points
in the momentum fraction $x$ and $N_{\rm pdf}$ is the number of total independent
flavors.
%
Then, a $N_xN_{\rm pdf}\times N_xN_{\rm pdf}$ covariance matrix on all
those PDF values can be constructed from the MC replicas,
\begin{equation}
{\rm cov}_{ll'}=\frac{1}{N_{\rm rep}-1}\sum_{k=1}^{N_{\rm rep}}X_{lk}X^T_{kl'}=\frac{1}{N_{\rm rep}-1}XX^T \, ,
\end{equation}
where $X_{lk}$ is the PDF value on $l$-th grid point given by $k$-th MC replica subtracted
with the corresponding value from central PDF set,
$N_{\rm rep}$ is the total number of MC replicas.
The above covariance matrix can be rewritten in its singular value decomposition
form     as follows,                               
\begin{equation}
{\rm cov}_{ll'}=\frac{1}{N_{\rm rep}-1}(USV^T)(USV^T)^T\, ,
\end{equation}
where $S$ is a diagonal matrix constructed out from singular values of $X$, $V$ is
an orthogonal $N_{\rm rep}\times N_{\rm rep}$ matrix of coefficients, and $U$ is
a $N_{x}N_{\rm pdf}\times N_{\rm rep}$ matrix containing orthogonal eigenvectors
of the covariance matrix with nonzero eigenvalues.
%
Indeed, the matrix $V$ gives
a Hessian basis built upon linear combinations of original MC replicas, which
reproduces fully the covariance matrix given by the original MC replicas.
%
This basis
can be further truncated using Principle Component Analysis with a certain error
metric, resulting in a smaller Hessian PDF set,
an approach similar to the case of the META-PDF method.

A third option is provided by the compressed Monte Carlo
(CMC) method~\cite{Carrazza:2015hva}. In this case,
an ensemble of pseudo-MC replica PDFs (CMC-PDFs) are generated. The CMC-PDFs
have a different statistical interpretation compared to the native MC PDFs. However, certain
statistical measures, such as the mean, covariance matrix, skewness, kurtosis and the
Kolmogorov distance  can be reconstructed in a similar way to the native MC PDFs.
%
The CMC-PDFs aim
to preserve some of the non-Gaussian features in the prior given by MC900 in addition
to the Gaussian features, for which the Hessian form is more adequate. Note that in the
current prescription some of the non-Gaussian behaviors from individual PDF sets have been
smoothed out due to the symmetric formula used in converting Hessian PDFs to MC replicas.
The compression starts with a figure of merit or error function ERF,
\begin{equation}
{\rm ERF}=\sum_k \frac{1}{N_k}\sum_i\left(\frac{C_i^{(k)}-O_i^{(k)}}{O_i^{(k)}}\right) \ ,
\end{equation}
where $k$ runs over the number of chosen statistical estimators, $N_k$ is a
normalization factor, $O_i^{(k)}$ is the value of the $k$-th estimator calculated
at the generic point $(x_i,Q_i)$ from the
prior and $C_i^{(k)}$ is the corresponding value of the same estimator in the
compressed set. The compressed set is simply a subset of the MC900 ensemble. For
any given number of total MC replicas, the compressed set is chosen by minimization
of the above error function using a genetic algorithm.   

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[t]
\begin{center}
  \includegraphics[scale=0.4]{4-fitting/plots/xg-mc900_vs_mch100_vs_meta30.pdf}
  \includegraphics[scale=0.4]{4-fitting/plots/xg-mc900_vs_cmc100.pdf}
  \includegraphics[scale=0.4]{4-fitting/plots/xdbar-mc900_vs_mch100_vs_meta30.pdf}
  \includegraphics[scale=0.4]{4-fitting/plots/xdbar-mc900_vs_cmc100.pdf}
   \caption{\small
   Comparison of the gluon and $\bar d$-quark PDFs at a scale of $Q^2=100\,{\rm GeV}^2$
   between the PDF4LHC15 prior and the two reduced Hessian sets, and between the prior and
   the compressed MC set, normalized to the central value of the prior~\cite{Butterworth:2015oua}. 
    \label{fig:comb1}
  }
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

These three compression and reduction techniques were used to construct the
three PDF sets that are the main output of the
2015 PDF4LHC recommendation: a Hessian set with $N_{\rm eig}=30$ eigenvectors
({\tt PDF4LHC15\_30}), a
Hessian set with  $N_{\rm eig}=100$ eigenvectors ({\tt PDF4LHC15\_100}), and a compressed MC set with
 $N_{\rm rep}=100$ replicas ({\tt PDF4LHC15\_mc}). All of these are constructed from the same prior
(MC900), but have a slightly different focus in each case.
%
The symmetric PDF uncertainties of any observables
can be calculated using Eq.~(\ref{eq:syhe}) for Hessian sets and the usual
master formula for MC PDFs, Eq.~(\ref{mastersig}).

Fig.~\ref{fig:comb1} shows the comparison of the
central value and the uncertainties of the gluon and $\bar d$ quark PDFs for
the PDF4LHC15 prior and the three reduced sets.
%
The agreement between the
Hessian set with 100 eigenvectors and the prior is good for all PDF combinations in the
complete range of $x$.
%
The Hessian set with 30 eigenvectors also shows good 
agreement with the prior in the $x$ range related to precision physics measurements,
but gives slightly smaller uncertainty in the extrapolation regions at small-
and large-$x$ as a tradeoff for the smaller number of PDF eigenvectors.
%
The compressed MC set also
agrees well with the prior in most
of the region for the mean and uncertainty except for small overall fluctuations.

The PDF4LHC Working Group recommendation for the usage of these different
PDF4LHC15 sets depends on the particular case under consideration:
\begin{itemize}
\item
Use individual PDF sets, and, in particular, as many of the modern PDF sets as possible,
for comparisons between data and theory for Standard Model measurements.

\item
Use the {\tt PDF4LHC15\_mc} sets for searches for BSM phenomena where
non--Gaussian behaviour could be important.

\item
Use the {\tt PDF4LHC15\_30} sets for calculation of PDF uncertainties in situations when
computational speed is needed, or a more limited number of error PDFs may be desirable.

\item
Use the {\tt PDF4LHC15\_100}
sets for calculation of PDF uncertainties in precision observables.
 
\end{itemize}
The cases listed above are not exclusive, and one or the other will be more
appropriate depending on the theoretical interpretation of a given experimental measurement.

In addition,  two further combined PDF4LHC15 sets are provided
with $\alpha_s(m_Z)=0.1165$ and $0.1195$,
in order to be able to estimate  the uncertainty due to $\delta \alpha_s$.
%
The corresponding
uncertainty at 68\% CL for a generic cross-section $\mathcal{F}$ is given by
\begin{equation}
\delta^{\alpha_s}\mathcal{F}=\frac{\mathcal{F}(\alpha_s=0.1195)-\mathcal{F}(\alpha_s=0.1165)}{2},
\end{equation}
where $\mathcal{F}(\alpha_s)$ is the value calculated using the PDF together with the 
hard matrix elements evaluated at that $\alpha_s$ value.
%
The combined PDF$+\alpha_s$
error is then computed by adding in quadrature the two sources of
uncertainty,
\begin{equation}
\delta^{{\rm PDF}+\alpha_s}\mathcal{F}=\sqrt{(\delta^{\rm PDF}\mathcal{F})^2
+(r\cdot\delta^{\alpha_s}\mathcal{F})^2},
\end{equation}
where the rescaling factor $r=1$ is recommended but can be varied according to
user's choice of uncertainty on $\alpha_s(m_Z)$, as compared to the range
$\delta\alpha_s=0.0015$ adopted as default.

It is also noted that the PDF4LHC15 PDF sets can be further reduced to compact
sets with around ten eigenvectors or less if the applications are restricted to a certain
group of observables, {\it e.g.}, the cross sections and distributions in Higgs boson
production at the LHC.
%
That can be achieved either through the data set diagonalization
method~\cite{Pumplin:2009nk,Gao:2014fma} or the singular value decomposition
method~\cite{Carrazza:2016wte}, the latter leading to the so-called
Specialized Minimal PDF (SM-PDF) sets.
