.. _method-bayes_calibration-posterior_stats-kl_divergence:

"""""""""""""
kl_divergence
"""""""""""""


Calculate the Kullback-Leibler Divergence between prior and posterior


.. toctree::
   :hidden:
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**Specification**

- *Alias:* None

- *Arguments:* None


**Description**


The Kullback-Leibler (KL) Divergence, also called the relative entropy,
provides a measure of the difference between two probability distributions.
By specifying ``kl_divergence``, the KL Divergence between the posterior
:math:`f(\boldsymbol{\theta} | \textbf{y}^{Data})`  and the prior
:math:`f(\boldsymbol{\theta})`  parameter distributions is calculated such that


.. math:: 

   D_{KL} = \int f(\boldsymbol{\theta} | \textbf{y}^{Data} ) \log
   \frac{ f(\boldsymbol{\theta} | \textbf{y}^{Data}) }{ f(\boldsymbol{\theta}) }
   d\boldsymbol{\theta}

This quantity can be interpreted as the amount of information gained about the
parameters during the Bayesian update.

*Expected Output*

If ``kl_divergence`` is specified, the calculated value will be reported to the
screen at the end of the calibration, following the sample statistics of the
response functions. Example output is given below.

*Additional Discussion*

The quantity calculated is a :math:`k` -nearest neighbor approximation of the
possibly multi-dimensional integral given above. Therefore, some applications
whose true KL Divergence is quite close to zero may report a negative KL
Divergence.



