Predicting Probability Distribution
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In this toolkit, a probability distribution is represented as a discretized cumulative distribution function (CDF).

Quantile Regression and Distribution Regression
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There are two types of discretization of a CDF :math:`\tau=F(y|x)`.
In quantile regression, we discretize a CDF along with :math:`\tau`-axis, and a CDF is represented as a set of values :math:`F^{-1}(\tau|x)` for various :math:`\tau`.
In distribution regression, we discretize a CDF along with :math:`y`-axis, and a CDF is represented as a set of values :math:`F(y|x)` for various :math:`y`.
In this toolkit, the "knots" of a CDF correspond to the vertices in the following figure.

.. image:: ./two_types_of_regression.png
  :width: 600


Package dqs.torch.distribution
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The classes in package dqs.torch.distribution are used to translate the output of a neural network to a CDF.
In quantile regression, a class in this package stores the quantile levels :math:`\{\tau_{i}\}` as parameters, and interpret the output of a neural network as values :math:`\{ F^{-1}(\tau_{i}|x) \}`.
In distribution regression, a class in this package stores the boundaries :math:`\{y_{i}\}` as parameters, and interpret the output of a neural network as values :math:`\{ F(y_{i}|x) \}`.