.. _likelihood:

===================
Likelihood function 
===================


Having defined a PDF model 
:math:`\mathbf{f}: \boldsymbol{\theta} \in \mathbb{R}^{N_{\rm params}} \to \mathbb{R}^{N_{\rm fl}\times N_{\rm x}}` 
(see the practical example in :ref:`Les Houches Tutorial <in_les_houches>`)
colibri allows us to sample the posterior distribution of its parameters :math:`\boldsymbol{\theta}`
given a :ref:`prior distribution <bayesian_prior>` :math:`\pi(\boldsymbol{\theta})`
and a likelihood function :math:`\mathcal{L}(\mathbf{D} | \boldsymbol{\theta})`.

In this section we will discuss the form of the likelihood function and its implementation in colibri,
note that a complementary discussion of the likelihood function 
can be found in the `NNPDF documentation <https://docs.nnpdf.science/figuresofmerit/index.html>`_

The basic form of the likelihood function used during sampling is a chi-squared function and is given by

.. math::
    \begin{align}
    \label{eq:likelihood}
    \mathcal{L}(\mathbf{D} | \boldsymbol{\theta}) &= (\mathbf{D} - FK[\mathbf{f}(\boldsymbol{\theta})])^T C_{\rm t_0}^{-1} (\mathbf{D} - FK[\mathbf{f}(\boldsymbol{\theta})]) \; ,
    \end{align}

where :math:`\mathbf{D}` is the data vector, :math:`FK[\mathbf{f}(\boldsymbol{\theta})]` is the forward model 
(described in detail in :ref:`Forward Model <theory_predictions>`), and :math:`C_{\rm t_0}` is the :math:`t_0` covariance matrix
used to avoid the d'Agostini bias when the data has multiplicative uncertainties 
as described in the `NNPDF documentation <https://docs.nnpdf.science/figuresofmerit/index.html>`_.

During a fit, it is also possible to impose positivity and integrability constraints on the PDFs.
This is achieved similarly to how it is done in the NNPDF framework, by adding extra Lagrange penalty terms to the likelihood function.

Positivity Constraints
^^^^^^^^^^^^^^^^^^^^^^
Positivity constraints are implemented in colibri by adding the following penalty term to the likelihood function

.. math::
    \begin{align}
    \label{eq:pos_constraint}
    \mathcal{L}(\mathbf{D} | \boldsymbol{\theta}) \to \mathcal{L}(\mathbf{D} | \boldsymbol{\theta}) + \Lambda_{\rm pos} \sum_{k}\sum_{i} Elu_{\alpha}(-\tilde{f}_k(x_i,Q^2)) ,
    \end{align}

where by defauly :math:`Q^2 = 5 GeV^2` and the :math:`n_i` values :math:`x_i` given by 10 points logarithmically spaced 
between :math:`5 \cdot 10^{−7}`  and :math:`10^{-1}` and 10 points linearly spaced between 0.1 and 0.9.

For a detailed description on how to include the positivity constraints in the likelihood function during a fit
see the tutorial ... (TODO a tutorial on fitting where positivity constraints are included).



Integrability Constraints
^^^^^^^^^^^^^^^^^^^^^^^^^
Integrability constraints are implemented in colibri by adding the following penalty term to the likelihood function

.. math::
    \begin{align}
    \label{eq:int_constraint}
    \mathcal{L}(\mathbf{D} | \boldsymbol{\theta}) \to \mathcal{L}(\mathbf{D} | \boldsymbol{\theta}) + \Lambda_{\rm int} \sum_{k}\sum_{i} \bigg[ x_i f_k(x_i, Q_0^2) \bigg]^2 ,
    \end{align}

where :math:`Q_0` is the parametrisation scale and the points :math:`x_i` run over a set of values in the small :math:`x`
region of the grid of the FK-table; in practice one often only takes the smallest :math:`x` value of the grid to 
enforce the condition.