.. _method-bayes_calibration-posterior_stats-kde:

"""
kde
"""


Calculate the Kernel Density Estimate of the posterior distribution


.. toctree::
   :hidden:
   :maxdepth: 1



**Specification**

- *Alias:* None

- *Arguments:* None


**Description**


A kernel density estimate (KDE) is a non-parametric, smooth approximation of
the probability density function of a random variable. It is calculated using
a set of samples of the random variable. If :math:`X`  is a univariate random
variable with unknown density :math:`f`  and independent and identically
distributed samples :math:`x_{1}, x_{2}, \ldots, x_{n}` , the KDE is given by

.. math:: 

   \hat{f} = \frac{1}{nh} \sum_{i = 1}^{n} K \left( \frac{x - x_{i}}{h} \right).

The kernel :math:`K`  is a non-negative function which integrates to one. Although
the kernel can take many forms, such as uniform or triangular, Dakota uses a
normal kernel. The bandwidth :math:`h`  is a smoothing parameter that should be
optimized. Choosing a large value of :math:`h`  yields a wide KDE with large
variance, while choosing a small value of :math:`h`  yields a choppy KDE with
large bias. Dakota approximates the bandwidth using Silverman's rule of thumb,

.. math:: 

   h = \hat{\sigma} \left( \frac{4}{3n} \right)^{1/5},

where :math:`\hat{\sigma}`  is the standard deviation of the sample set
:math:`\left\{ x_{i} \right\}` .

For multivariate cases, the random variables are treated as independent, and a
separate KDE is calculated for each.

*Expected Output*

If ``kde`` is specified, calculated values of :math:`\hat{f}`  will be output to
the file ``kde_posterior``.dat. Example output is given below.


