dqs.torch.loss.Brier
================================================

This class is used to compute the Brier score, 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 average squared difference 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 Brier score for an actual outcome :math:`y` can be calculated as

.. math::

   \sum_{i=1}^{n} (f[i] - b)^2,

where :math:`b` is the binary indicator if :math:`y=i`.  In other words, :math:`b=1` if :math:`y=i`, and :math:`b=0` otherwise.


.. code-block:: python

    class Brier(
        distribution,
        loss_boundaries)


Parameters:

=================== ================ ===================================================================================
Args                Type             Description
=================== ================ ===================================================================================
distribution        dqs.distribution Object from dqs.distribution package to represent probability distribution.
boundaries          list (float)     Boundaries used to compute the Brier score.
=================== ================ ===================================================================================


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 the Brier score based on ``pred`` and labels ``y``.

.. code-block:: python

    boundaries = torch.linspace(0.0, 10.0, 11)
    dist = dqs.distribution.DistributionLinear(boundaries)
    loss_fn = dqs.loss.Brier(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)