dqs.torch.loss.NegativeLogLikelihood
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This class is used to compute the negative log-likelihood, which is one of the measures of the accuracy of probabilistic predictions.
It is calculated by comparing the predicted probability distribution to the actual outcome. 
It quantifies the negative log-likelihood between the predicted probability distribution and the corresponding actual outcome.

Suppose that the predicted probability distribution is represented as a CDF with :math:`n` knots, and the probability that the target value is contained in the :math:`i`-th bin can be represented as ``f[i]``.
Then the negative log-likelihood for an actual outcome :math:`y` can be calculated as

.. math::

   - \log f[i],

where :math:`i` is the index of the bin :math:`y` is contained.


.. code-block:: python

    class NegativeLogLikelihood(
        distribution,
        loss_boundaries)


Parameters:

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Args                Type             Description
=================== ================ ===================================================================================
distribution        dqs.distribution Object from dqs.distribution package to represent probability distribution.
boundaries          list (float)     Boundaries used in negative log-likelihood.
=================== ================ ===================================================================================


loss(pred, y, e=None)
---------------------------------

Parameters:

=================== ================ ===================================================================================
Args                Type             Description
=================== ================ ===================================================================================
pred                Tensor (float)   Estimated probability distribution to be evaluated.
y                   Tensor (float)   One-dimensional tensor to represent labels from a dataset.
e                   Tensor (bool)    Optional parameter for survival analysis.  One-dimensional tensor to represent censored (False) or uncensored (True).
=================== ================ ===================================================================================

Return type: Tensor representing a single float.


Example
---------------------------------

The following code computes negative log-likelihood based on ``pred`` and ``y``.

.. code-block:: python

    boundaries = torch.linspace(0.0, 10.0, 11)
    dist = dqs.distribution.DistributionLinear(boundaries)
    loss_fn = dqs.loss.NegativeLogLikelihood(dist, boundaries)
    pred = torch.Tensor([[0.4,0.6],[0.2,0.8]])
    y = torch.Tensor([5.0,5.0])
    loss = loss_fn.loss(pred, y)